GOODRICH'S 
ANALYTICAL  HARMONY. 


A  THEORY  OF 

MUSICAL  COMPOSITION  FROM  THE 
COMPOSER'S  STANDPOINT. 


INTRODUCING  AN  EXPLANATORY  TREATISE  UPON  UNRELATED  TONES:  A  NEW 
SYSTEM  OF  HARMONIC  COUNTERPOINT  AND  DIAGRAM  ILLUSTRA- 
TIONS OF  MUSICAL  FORM  AND  CONSTRUCTION. 


BY  A.  J.  GOODRICH, 

AUTHOR  OF 

"COMPLETE   MUSICAL  ANALYSIS,"  "MUSIC  AS  A 
LANGUAGE,"  ETC. 


PUBLISHED  BY 

THE  JOHN  CHURCH  COMPANY. 

CINCINNATI.         NEW  YORK.         CHICAGO. 


(  OPYRIGIHED,  1893,   BY   THK  JO1IS  CllVK.  II    CO. 

All  Rights  Reservc.l. 


Muaic 
Library 


PREFACE. 


*"pHE  advantages  accruing  from  a  knowledge  of  Harmony  are  not 
sufficiently  understood,  except  by  those  who  are  ambitious  to 
compose.  Every  singer,  performer,  teacher  and  critic  is  benefited 
in  knowing  the  principles  of  chord  succession,  harmonization,  etc. 
Pianists  who  possess  this  information  have  an  immense  advantage 
in  the  knowledge  that  modulatory  tones,  suspensions  and  appoggia- 
turas  are  accented  ;  that  dissonances  are  to  be  connected  with  the 
consonances  to  which  they  resolve  ;  that  passing  tones  are  unac- 
cented; that  anticipations  are  slightly  marked,  and  that  different 
kinds  of  cadences  require  different  kinds  of  punctuation.  As  an 
aid  to  sight-reading  (that  most  necessary  accomplishment)  a  knowl- 
edge of  Harmony  is  indispensable,  for  it  enables  one  to  anticipate 
a  considerable  portion  of  music  by  being  familiar  with  the  notation, 
resolution  and  progression  of  chords  in  general. 

Our  present  system  of  music  has  been  gradually  evolved  during 
centuries  of  artistic  and  scientific  progress.  Some  of  the  world's 
greatest  geniuses  laid  the  foundation,  built  up  the  structure  and 
added  the  ornamentation.  The  theorist,  has,  therefore,  but  little 
to  do  beyond  that  of  presenting  the  material  of  composition  and 
showing  how  this  has  been  employed.  Certain  principles  and 
theories  may  be  deduced  from  the  music  of  a  Beethoven,  and  these 
are  to  be  systemized  and  explained.  But  while  the  creative  impulse 
in  music  continues  to  manifest  itself  it  must  be  unfettered  by  arbi- 
trary rules  and  prohibitions.  Recent  composers,  in  their  use  of 
harmony,  have  gone  far  beyond  the  formulas  and  precepts  of  text- 
books. It  is  no  longer  possible,  according  to  existing  systems  of 
theory,  or  of  acoustics,  to  explain  the  harmonic  structure  of  such 
works  as  Saint-Saens'  Danse  Macabre,  Grieg's  Norwegian  Dances, 
Tschaikowski's  Francesca  da  Rimini  ',  or  the  later  music-dramas 

3 


4  I'KKI-ACE. 

of  Wagner.  These  creators  of  music  followed  a  higher  law  than 
didactic  theorem,  and  the  theorist  should  act  only  intermediately, 
explaining  to  the  student  the  artistic  phenomena  of  cause  and 
effect. 

During  the  past  twenty-four  jrears  since  this  system  was  com- 
menced the  author  has  confined  himself  principally  to  this  task  : 
i.  To  present  the  material  and  technic  of  composition  in  system- 
atic and  graded  order.  2.  To  explain  this  analytically  and  clearly. 
3.  To  illustrate  the  application  of  this  material  in  the  construction 
of  music.  4.  To  show  the  esthetic  effect  (and,  consequently,  the 
object)  of  certain  chords  and  progressions.  These  are  the  main 
features  of  the  present  system,  which  is  based  upon  the  actual 
results  of  composition  rather  than  upon  existing  theoretical  works ; 
and  whatever  merits  this  book  may  possess  are  thus  ascribed  to  the 
influence  of  Scarlatti,  Couperin,  Bach,  Mozart,  Beethoven,  Schubert, 
Schumann,  Chopin,  Mendelssohn,  Wagner,  Rubinstein,  Dvorak, 
Gounod,  Saint-Saens,  Jensen,  L,assen,  Goldmark,  Tschaikowski, 
Grieg,  and  Mascagni,  not  to  Zarlino,  Rameau,  Kirnberger,  Gott- 
fried Weber,  Marx,  Weitzmaun,  Richter,  uor  Riemauu. 


TO  THE  TEACHER. 


1YJOT  only  is  the  plan  of  this  book  different  from  that  of  other 
harmony  books,  but  some  of  the  current  nomenclature  has  been 
rejected.  This  need  not  occasion  confusion,  for  the  old  and  the 
new  names  are  mentioned  synonymously.  One  instance  is  the  5th 
of  a  normal  scale.  This  interval  has  heretofore  been  called :  Pure, 
Major,  Standard,  and  Perfect.  Our  tempered  fifths  are  neither  pure 
nor  perfect,  therefore  these  names  are  inappropriate.  Major  is  mis- 
leading, because  major  can  not  consistently  be  applied  to  such 
intervals  as  the  normal  4th  and  5th,  which  are  the  same  in  both 
major  and  minor  scales.  The  word  standard  is  more  acceptable, 
though  the  author  prefers  calling  the  4th  and  the  5th  of  every  nor- 
mal scale  Normal  intervals. 

A  few  words  of  explanation  are  also  offered  in  reference  to  the 
designation  of  voice-parts.  In  the  elemental  chapters  of  this  book 
chords  appear  in  their  close  positions — three  parts  in  the  treble  and 
one  in  the  base.  As  it  is  essential  to  correct  chord  progression 
that  the  student  should  follow  the  movement  of  each  part,  it  has 
been  deemed  advisable  to  give  names  to  these  parts  corresponding 
to  the  voices  which  would  sing  them  if  they  were  vocal.  Therefore 
tli2  author  has  named  the  different  parts  of  such  chords  as  these : 
Soprano,  the  highest ;  mezzo-soprano,  the  middle 
r£^nii3±q  part;  contralto,  the  lowest  of  the  three  treble  parts. 
Y-fpr)'^1*?  & — I  This  nomenclature  is  adopted  for  ail  such  chords, 
which  sound  exactly  as  written.  The  word  "  tenor" 
is  not  here  applied  to  any  part  of  a  chord  in  close  position,  because 
a  tenor  part  when  sung  from  the  treble  staff  sounds  an  octave  lower 
than  written.  In  the  chapter  on  Harmonic  Counterpoint  (where 
dispersed  harmony  is  first  introduced)  the  original  mezzo-soprano 
part  becomes  tenor  whenever  that  part  is  to  be  inverted  in  order 

5 


6  TO   THE   TEACHER. 

to  form  open  harmony.  It  is  customary  to  name  the  lowest  treble 
part  tenor;  but  an  equal  distribution  of  the  voices  requires  that  all 
the  parts  shall  be  in  open  position,  and  this  compels  us  to  choose 
the  middle  upper  part  for  inversion.  The  following  examples  in 
notation  illustrate  these  different  methods : 

b       e  At  (a)  the  chord  appears  in  close  position. 

At  (6)  the  lowest  treble  part  of  the  initial 
chord  is  considered  as  tenor  and  inverted. 
At  (f)  the  tenor  part  is  an  inversion  of  the 


& 


m 

-so 1 


_  _ 

|Q^  —  ts-^  :g=  -•&—       middle  treble  part  in  Ex.  (a).     Any  one  who 
prefers   the  arrangement   at   (b}   must   have 


queer  notions  about  vocal  music  and  dispersed  harmony. 

Particular  attention  is  directed  to  the  necessity  of  transposing 
the  exercises  into  a  variety  of  scales,  for  this  is  the  surest  way  to 
a  mastery  of  the  different  subjects.  One  of  our  most  accomplished 
harmony  teachers  has  noted  with  satisfaction  this  leature  o!  the 
work.  After  the  student  has  completed  a  certain  harmonization 
the  act  of  transposition  should  apply,  usually,  to  the  melody  only. 
Then  the  harmonization  is  to  be  completed  in  the  new  scale.  This 
is  more  beneficial  than  to  transpose  both  melody  and  harmony. 

By  closing  the  books  a  class  may  write  on  a  blackboard  the 
various  exercises  without  fear  of  unduly  referring  to  the  solutions 
in  the  text.  Those  who  pursue  the  study  without  a  tutor  will 
understand  that  if  they  consult  these  solutions  before  the  examples 
have  been  worked  out  they  will  acquire  only  a  superficial  knowl- 
edge of  the  subject. 

Throughout  the  text  ellipsis  *  •>-  appear  whenever  an 

example  is  to  be  worked  out  before  proceeding  farther. 

The  first  twenty  chapters  are  confined  to  concords,  that  the 
student  may  more  readily  learn  to  manage  chords  without  being 
burdened  with  the  additional  rules  of  resolution  which  apply  to 
discords.  No  discord  is  introduced  until  it  is  required,  and  in  this 
way  one  subject  leads  to  another. 

Inverted  bases,  being  rather  difficult  of  management,  are  not 
permitted  before  the  2jih  chapter.  For  a  similar  reason  the  intro- 
duction of  open  harmony  is  postponed  until  the  latter  part  of  the 
book.  The  teacher's  attention  is  therefore  directed  to  the  sequence 
of  subjects  as  set  forth  in  the  Arrangement  of  Contents,  and,  ex- 
cepting for  a  particular  purpose,  the  author  would  not  advise  any 
deviation  from  this  order.  The  present  Arrangement  of  Contents 


TO  THE   TEACHER.  7 

has  occasioned  more  anxiety  and  thought  than  any  other  feature  of 
the  work,  and  a  comparison  with  any  standard  treatise  on  Harmony 
will  show  great  dissimilarity  in  this  respect. 

The  author's  plan  embraced  an  exhaustive  chapter  on  the  Pro- 
gressive Development  of  Harmony  during  different  Epochs  and  the 
tendency  of  recent  chord  combinations,  together  with  a  list  of  Refer- 
ences and  a  somewhat  discursive  chapter  on  the  Supposed  Physical 
Basis  of  Harmony.  Owing  to  the  already  considerable  size  of  the 
book  the  publishers  advised  that  those  chapters  be  omitted  from 
the  present  edition  and  issued  separately  at  some  future  time.  As 
these  parts  are  not  absolutely  essential  to  a  text-book  the  suggestion 
lias  been  adopted. 

It  is  scarcely  necessary  to  add  that  the  chapters  do  not  represent 
the  number  of  lessons.  Certain  chapters,  such  as  XIII,  XIV, 
XXXV,  XL,  may  require  three  or  four  lessons  for  their  thorough 
comprehension. 

A.  J.  GOODRICH. 


CONTENTS. 


PAGE 

PREFACE 3 

To  THE  TEACHER 5 

PART  I. 

CHAPTER  i.  Natural  Intervals  of  the  Major  Scale  ....  13 

CHAPTER  2.  "                   "                "         Minor       "        ....  17 

CHAPTER  3.    Formation  of  Major  Concords 19 

CHAPTER  4.  "  "    Minor  "  23 

CHAPTER  5.  Major  and  Minor  Concords  Re-arranged    .  26 

CHAPTER  6.    Harmonization  in  Three  Parts 28 

PART  II. 

CHAPTER  7.  Theory  of  Harmonic  Progression.  Chord  Suc- 
cessions Re-arranged '. 32 

CHAPTER  8.  Harmonic  Progression  in  Four  Parts.  Addi- 
tion of  the  Fundamental  Base 35 

CHAPTER  9.  Harmonization  of  a  Given  Theme.  Application 

of  Theoretical  Principles 38 

CHAPTER  10.    Harmonization  of  Melodic  Skips  of  a  Third       42 

CHAPTER  n.  "  "  "  "  Fourth       44 

PART  111. 

CHAPTER  12.  Forbidden  Progressions 47 

CHAPTER  13.  Thirty  Harmonic  Progressions  in  a  Major 

Key.  With  and  Without  Connecting  Notes  ...  51 
CHAPTER  14.  Harmonization  of  Themes,  with  and  Without 

Connecting  Notes.  Chord  Relations 55 

CHAPTER  15.  Another  Method  of  Harmonizing  Skips  of 

a  Third     60 

PART  IV. 

CHAPTER  16.  The  Harmonic  Minor  Scale  and  its  Con- 
sonant Triads 64 

CHAPTER  17.     The   Major  and    Minor   Modes  Combined. 

Introduction  to  Modulation    .    .  66 


10 


CONTENTS. 


CHAPTER  18.  Primary  Modulations  to  all  Related  Keys 

excepting  the  Subdominant  69 

CHAPTER  19.  Themes  for  Harmonization  Illustrating  the 

Preceding  Modulations 75 

CHAPTER  20.  Modulations  from  the  Minor  Mode,  with 
Illustrative  Themes 


PART  V. 

CHAPTER  21.  Formation  and  Resolution  of  the  Dominant 
7th  Chord 

CHAPTER  22.  Omission  of  the  Third  or  Fifth  from  the 
Dominant  7th  Chord  ...  

CHAPTER  23.  Major  and  Minor  Resolutions  of  the  Domi- 
nant 7th  Chord.  Illustrative  Themes  .... 

CHAPTER  24.  Four  Resolutions  of  the  Dominant  7th 
Chord  Analyzed  

CHAPTER  25.  The  Preceding  Resolutions  Classified  and 
Characterized.  Direct  and  Avoided  Cadences 

CHAPTER  26.    Avoided  Cadences  Illustrated  .      

PART  VI. 


84 
89 
92 

95 
98 


CHAPTER  27.    Inverted  Bases.    Their  Object  and  Effect  104 

CHAPTER  28.    Unrulable  Progressions  and  Resolutions  113 
CHAPTER  29.    Dissonant  Triads  — Imperfect,  Augmented 

and  Diminished 117 

PART  VII. 

CHAPTER  30.  Origin  and  Principal  Resolution  of  the 

Diminished  7th  Chord 126 

CHAPTER  31.  Natural  Modulations  to  the  Related  Minor 
Keys  by  means  of  the  Diminished  7th 
Chord.  Another  Mode  of  Transition  .  .  .  130 

CHAPTER  32.  Diminished  and  Corresponding  Dominant 

7th  Chords 133 

CHAPTER  33.  The  Diminished  7th  Chord  Inverted.  Ap- 
plication of  the  Corresponding  Domi- 
nant 7th.  Intermediate  and  Terminal 
Resolutions 135 

CHAPTER  34.  Themes  Illustrating  the  Preceding.  Far- 
ther View  of  Inverted  Bases 139 

PART  VIII. 

CHAPTER  35.    Principal   and    Secondary   7th    Discords. 

Their  Origin,  Application  and  Effect  .  .       14 


CONTENTS. 


II 


CHAPTER  36.    Additional  Chord  Progressions 151 

CHAPTER  37.  Succession  of  Dominant  7th  Chords  Dia- 
tonicallyand  Chromatically.  Another  Means 
of  Transition 158 

PART  IX. 

CHAPTER  38-  Fifteen  Enharmonic  Transitions  by  means 
of  Three  Primary  Diminished  7th  Chords. 
Theory  of  Notation 164 

CHAPTER  39-  The  Diminished  7th  Chord  as  a  means  of 
Enharmonic  Transition.  Chromatic  Har- 
monization    168 

CHAPTER  40.  The  Diminished  7th  Chord  as  a  Passing 
Harmony  to  the  Tonic  and  to  the  Domi- 
nant 7th 175 

CHAPTER  41.  Harmonic  Cadences  in  Major:  i.  Authentic. 
2.  Complete.  3.  Perfect.  4.  Extended-Perfect. 
5.  Avoided.  6.  Deceptive.  7.  Incomplete.  8.  After  183 

CHAPTER  42.  Harmonic  Cadences  in  Minor:  i.  Authentic. 
2.  Complete.  3.  Perfect.  4.  Extended-Perfect. 
5.  Avoided.  6.  Deceptive.  7.  Incomplete.  8.  After. 
9.  Ambiguous  192 

PART  X. 

CHAPTER  43.    Augmented  6th  Chords.    Their  Derivation, 

Application  and  Effect,  No.  1 199 

CHAPTER  44.    Augmented  6th  Chords  Continued,  No,  2  .  205 

CHAPTER  45.    Augmented  6th  Chords  Continued,  No.  3  .  209 
CHAPTER  46.    Application  of  the  Various  Augmented  6th 

Chords  in  Harmonization  Concluded  .  .  212 


PART  XI. 

CHAPTER  47.  Harmonic  Progressions  in  General.  Their 

Esthetic  Effect 220 

CHAPTER  48.    Figured  Bases.    A  Cursory  View 228 

CHAPTER  49.  The  Natural  and  Melodic  Minor  Scales. 

Their  Harmonies 231 

CHAPTER  50.    Principal  and  Secondary  9th  Chords    .  • '.  237 

PART  XII. 

CHAPTER  51.    Suspension.    The  Theory  Illustrated     ...  242 

CHAPTER  52.    Suspension  Continued        249 

CHAPTER  53.    Pedal-Note.     (Organ-Point) 255 

CHAPTER  54.    Seven  Additional  Resolutions  of  the  Domi- 
nant 7th  Chord 263 


12 


CONTENTS. 
PART  XIII. 


CHAPTER  55.    Duplication  and  Omission 268 

CHAPTER  56.  Related  and  Unrelated  Tones:  'Harmonic, 
Passing,  Appoggiatura,  Suspension,  Anticipation, 

Stationary  Tone  and  Embellishment      276 

CHAPTER  57.    Related  and  Unrelated  Tones  Continued. 

Illustrative  Themes 283 

PART  XIV. 

CHAPTER  58.  Harmonic  Counterpoint.     Elementary   Species  291 

CHAPTER  59.               "                              "                     Compound  296 

CHAPTER  60.               "                              "                     Ornate  305 

CHAPTER  61.  Harmonic  Accompaniment  Illustrated    .  .  315 

PART  XV. 

CHAPTER  62.    Interdicted  Progressions.    False  Relation  .  .  .  325 

CHAPTER  63.    Analysis  of  Harmonic  Sequence 333 

CHAPTER  64.    Influence  of   Rhythm   and    Phrasing  upon 

Harmonic  Movement 337 

PART  XVI. 

CHAPTER  65.  Harmony  in  Five,  Six,  Seven,  Eight,  Nine 

and  Ten  Parts  342 

CHAPTER  66.  Abrupt,  Enharmonic  and  Remote  Transi- 
tion. Key  and  Chord  Relations  .  348 

CHAPTER  67.    Altered  Discords.     Double  and  Triple  Dissonances      356 


PART  XVII. 
CHAPTER  68.    Musical  Form  and  Construction.    Elementary 


Models  and  Forms 


;68 


CHAPTER  69.    Musical  Form  and  Construction  Continued. 

Rhythm .  .  373 

CHAPTER  70.  Musical  Form  and  Construction  Concluded. 

The  Sonata  Form  in  Major  and  in  Minor:     Tonality, 

Development,    Form-Diagrams,   Affinity  of  Motives. 

Conclusion 376 

INDEX  OF  SUBJECTS 389 

KEY  TO  EXAMPLES 393 


GooDRicH's  ANALYTICAL  HARMONY, 


PART    I. 


Chapter  I. 


NATURAL  INTERVALS  OF  THE  MAJOR  SCALE. 

A  FTER  countless  experiments  in  scale  construction,  and  centuries 
£*•  of  progressive  development  in  musical  theory  and  practice, 
modern  composers  have  uniformly  adopted  what  is  called  the  Nor- 
mal major  scale,  as  the  most  natural  and  important  series  of  single 
tones  proceeding  from  and  returning  to  a  given  tonic  or  key-tone. 

As  the  scale  is  the  foundation  of  all  serious  music  study,  both 
theoretical  and  practical,  a  knowledge  of  scale  construction  is  here 
presupposed.  Not  every  teacher,  however,  appreciates  the  necessity 
of  a  thorough  and  intimate  acquaintance  with  scales  in  all  keys 
and  all  forms.  Suppose,  for  instance,  this  fragment  of  melody 


should  «appear: 


Ex.    i. 


In  order  to  an- 


alyze, transpose,  :>r  accompany  these  tones  we  must  know  that  they 
belong  essentially  to  the  scale  of  A-flat,  and  that  A-flat  is  the  key- 
tone.  The  e-flat  and  d-flat  presuppose  b-flat  and  a-flat,  and  these, 
together  with  c,  g,  and/,  constitute  the  scale  of  A-flat. 

interval  is  time  between  events,  or  space  between  things.  \Yith 
the  scale  as  a  basis  the  staff  degrees  afford  the  simplest  means  of 
enumerating  intervals.  In  musical  theory,  interval  refers  to  die 


14  GOODRICH'S  ANALYTICAL,  HARMON v. 

distance  between  tones,  reckoned  either  up  or  down.      Two  pre- 
liminary explanations  are  necessary: 

1.  The  foundation  tone  is  counted  one. 

2.  The  numerical  terms   2d,  3d,  4th,  and  so  on.  are  not  to  be 
understood  as  fractions  of  a  whole,  but  as  expressing  the  number 
of  degrees  included  in  a  certain  interval.     For  instance,  this  is  a 


5th:     Ex-  2-  F/W— {"-—      because  g  is  the  fifth  natural  tone  in  the 


scale  of  c :     Ex"  3"  Fffc    ^     f-  The  black  notes  show  that 

"i      a    a    4    5 

live  degrees  of  the  staff  are  involved  in  ascertaining  the  numerical 
distance  from  c  (i)  to  g  (5). 

Intervals   are   computed    both    melodically    and    harmonically. 
When  they  occur  separately  they  are  melodic.     Thus  in  singing 


i     a 


the  voice    ascends  a   3d,   and    in    this : 


it  descends  a  5th.     In  both  instances  the  first 


tone  is  counted  one. 


When  the  tones  occur  simultaneously :    Ex-  6-  F/fczl 


the 


interval  is  harmonic;  but  it  is  counted  in  the  same  manner  as 
was  Ex.  4.  When  three  notes  appear  simultaneously  they  involve 
two  intervals.  These  may  be  computed  either  fundamentally  or 


componently :     Ex-  7  Ffe_  According  to  the  former  method 


the  interval  from  C  to  e  is  a  major  3d,  and  that  from  C  to  g  is  a 
normal  5th.  According  to  the  latter  there  is  a  major  3d,  c  to  e, 
and  a  minor  3d,  e  to  g.  The  results  are  the  same,  though  both 
methods  must  be  understood. 

In  order  that  this  melodic  motive :    Ex>  8-  rfft)         -I "  -J 

^/        ~~0       0       t 

be  transposed  to  any  key,  it  should  be  desc'.ibed  as  beginning  upon 
the  tonic  and  ascending  two  whole  steps.  The  result  in 


would  be  this :     Ex-  9- 


GOODRICH  S   ANALYTICAL    HARMONY. 


To  return  to  the  scale :  This  consists  of  two  equal  parts  called 
by  the  Greeks  tetrachords.  Each  tetrachord  contains  two  whole 
steps  and  one  half  step.  Marx,  following  the  Greek  theory,  called 
the  tonic  a  "  central  tone  "  around  which  the  other  tones  revolve, 
thus: 


I  !>!><•  i-  hair. 


Ex    10 


•  raftr-j  •  '  '[?'•- 

:^-q= 

| 

—  1  

r-«p  --—  '  i  i   i   H   i   *~ 

3=3=j£: 

Lower  list 

^f 

I—  ^     • 

(5*  •    ••' 
<J/ 

But  in  modern  practice  the  lower  half  should  appear  as  upper  half 
in  order  to  complete  the  scale : 


Ex.  ii. 


Observe  that  the  last  interval  in  each  tetrachord  is  a  minor  2d. 
The  intervals  of  this  scale  may  be  counted  either  melodically  or 
fundamentally,  as  was  the  chord  in  Ex.  7.  As  the  scale  is  alphabet- 
ical it  necessarily  proceeds  by  seconds,  as  from  a  to  b,  b  to  c-sharp, 
and  so  on.  From  a  fundamental  standpoint  we  have  a  tonic  (or 
prime),  a  2d,  3d,  4th,  5th,  6th,  yth,  and  octave.  The  first  seven  tones 
complete  the  scale,  since  8  is  a  duplicate  of  i.  Both  are  called  by 
the  letter  a,  and  thus  the  scale  might  be  continued  one  or  more 
octaves  without  altering  its  character.  If  the  scale  is  counted  back- 
ward the  general  result  is  the  same,  for  the  relationship  of  each  tone 
to  the  tonic  is  not  changed  by  being  reversed. 

This  appears  in  applying  the  vocal  syllables  to  the  scale  ascend- 
ing and  descending,  thus : 


Ex.   12. ' 


adlfe   ! 

1         3         4 

5 

S3         "*?       "*"     ^       £? 

5 

4        3 

a       i 

VL.     C 

—  ~  — 

^-j 

f^   <      &       '. 

,  

do     re     mi     fa     sol       la 


do     si      la    sol      fa    mi      re     do 


To  the  singer  the  tonic  is  always  do  (according  to  the  movable-Do 
system),  the  3d  is  always  mi,  the  5th  sol,  and  so  on.  This  much 
forms  the  basis  of  the  study  of  intervals. 

The  next  step  is  to  ascertain  a  more  specific  designation  for 
these  intervals  than  the  mere  numerical  terms  2d,  3d,  4th,  etc. 
It  is  not  sufficient  to  know  that  from  a  to  c  is  a  5th  ;  one  must  know 
what  kind  of  a  5th  this  is.  The  interval  known  in  scale  formation 


i6 


GOODRICH'S  ANALYTICAL  HARMONY. 


as  a  whole  step,  as  from  a  to  b,  is  called  theoretically  a  major  (large) 
2d.  This  applies  to  successive  degrees  when  there  is  a  chromatic 
tone  between  them,  thus  : 


Ex.  13. 


A  minor  2d  has  no  intermediate  tone.  These  small  seconds  are 
often  called  "  semi-tones,"  though  the  latter  expression  is  incorrect. 
In  all  major  scales  the  intervals  from  3  to  4  and  from  7  to  8  are 
minor  seconds.  If  the  numbers  were  reversed  the  intervals  would 
of  course  remain  the  same.  Accordingly  the  major'  scale  consists 
melodically  of  major  and  minor  seconds,  distributed  in  the  manner 
indicated  in  Ex.  n. 

The  intervals  of  the  scale  will  now  be  illustrated  ard  enumerated 
fundamentally. 

Major  2d.          Major  3d.       Nor.  4th.        Nor.  sth.        Maj.  6th.       Maj.  7th. 


G 


^ 


The  major  2d  has  been  described  as  a  whole  step.  The  major  3d 
is  generally  distinguished  as  containing  two  whole  steps,  but  a  sim- 
pler method  is  this :  the  3d  tone  in  every  major  scale  is  a.  major  3d 
from  the  tonic.  The  4th  and  5th  are  called  normal,  because  they 
are  the  same  in  both  normal  modes  (major  and  minor)  and  remain 
the  same  through  inversion,  as  here  : 


Ex.  15. 


Nor.  4th.      Nor.  sth.  Nor.  sth.  Nor.  4th. 


Counting  from  the  tonic,  d  is  a  4th  above  or  a  5th  below,  apd  c  is 
a  5th  above  or  a  4th  below. 

Mathematically  these  intervals  are  more  nearly  in  perfect  tune 
than  the  others,  and  in  actual  composition  the}'  are  treated  as  nor- 
mal. With  exception  of  the  normal  4th,  normal  5th,  and  perfect 
octave,  all  the  intervals  change  their  nature  through  the  process  of 
inversion : 


Ex.  16. 


Maj.  2d.          Mi.  7th.  Maj.  3d.  Mi.  6th.    Mi.  zd.    Maj.  7th.  Mi.  3d.  Maj.  6. 


GOODRICH  S    ANALYTICAL    HARMONY. 


The  6th  tone  in  a  major  scale  is  major,  and  so  is  the  jih,  the 
latter  being  only  a  minor  2d  below  the  octave.  Therefore  the 
simplest  way  of  naming  the  intervals  in  a  major  scale  is  this: 
The  2d,  3d,  6th,  and  yth  are  major;  the  4th  and  5th  are  normal. 
If  a  pupil  knows  the  scales  it  is  easy  to  tell  the  name  of  an  inter- 
val without  committing  to  memory  the  old  formulas, "  a  major  3d 
contains  four  semi-tones,"  and  others  in  proportion.  Examples 
similar  to  No.  14  should  be  written  in  all  major  scales  and  named 
by  the  student.  *  *  * 

These  additional  intervals  are  to  be  designated : 


Ex.  17 


->»• 


^HiiBi^^^=ls3^ 


^q 


Chapter  II. 


NATURAL  INTERVALS  OF  THE  MINOR  SCALE. 


MINOR   intervals  are  one  chromatic  step  smaller  than  major 
intervals.     C  and  e  constitute  a  major  third.     By  raising  c, 
or  lowering  e,  a  minor  third  will  result. 


Ex.  18. 


Maj.  3d.     Mi.  3d.      Mi.  3d. 


These  are  precise,  theoretical  distinctions,  the  same  staff  degrees 
being  employed.  In  each  instance  the  interval  is  a  3d ;  but  the  first 
is  large,  the  other  two  are  small. 

The  first  seven  natural  intervals  in  A-minor  should  now'  be  writ- 
ten and  named  theoretically,  according  to  directions : 


Those  intervals  not  affected  by  the  change  in  signature  retain 
the  same  names  as  in  Ex.  14.     Those  that  are  one  chromatic  step 


i8 


GOODRICH'S  ANALYTICAL  HARMONY. 


smaller  here  than  they  were  in  A-major  are  to  be  marked  minor 
(less).*  The  difference,  therefore,  between  major  and  minor  lies  in 
the  3d,  6th,  and  yth,  according  to  the  signature  of  each  mode. 
If  the  yth  of  a  minor  scale  is  raised,  it  will,  of  course,  become 
major,  but  the  natural  minor  scale  is  used  here  merely  to  show 
the  difference  between  major  and  minor  intervals  and  the  circum- 
stances under  which  they  occur.  The  2d,  4th,  and  5th  are  the 
same  in  both  modes.  The  3d,  6th,  and  yth  are  major  in  a  major 
scale,  and  minor  in  a  minor  scale. 

Two  comparative  tables  of  intervals  are  presented,  representing 
the  tonic  major  and  tonic  minor  of  G. 


Ex.  20. 


These  intervals  are  to  be  named  by  the  student. 

The  intervals  thus  far  employed  are  the  major  and  minor  2d, 
major  and  minor  3d,  normal  4th  and  5th,  major  and  minor  6th, 
major  and  minor  yth,  and  perfect  octave.  The  others  have  been 
purposely  omitted  until  they  shall  be  required. 


n 

a. 

1  «"  1 

r  —  *~~\ 

*?-, 

I    —    2 

1    3 

.gy.    . 
1    4 

1    —   5 

1  —  6 

r&  \ 
1  —  7 
_^_1 

•rfe-  —  -&  °-  — 

-«>  

-«'  

-&  

-«?  \ 

Maj.  2d. 


Mi.  3d. 


Nor.  4th.        Nor.  sth.        Mi.  6th. 


Mi.  yth. 


ji  L 

1 

fe-    . 

„ 

«  — 

1  °-  — 

* 

—  —  1 

Ex.  21. 


Similar  examples  should  be  written  in  other  natural  minor  scales. 

Theory  seems  to  differ  from  practice  in  the  designation  of  cer- 
tain intervals.  For  instance,  f  to  a-flat  is  a  minor  3d,  whereas  f 
to  g-sharp  is  an  augmented  2d.  On  all  keyed  instruments  a-JJat 
and  g-sharp  look  and  sound  alike ;  yet  there  is  a  considerable  differ- 
ence theoretically,  and  a  slight  difference  mathematically.  The 
vibration  numbers  are  in  the  following  proportions:  G-sharp^ 
412^;  a-flat,  422! .  In  theory, /to  g-sharp  is  an  augmented  2d, 
while  /  to  a-flat  is  a  minor  3d.  The  usual  difference  in  their 
resolutions  may  be  observed  by  comparing  (a)  with  (b). 


Ex.  22. 


natural  serves  as  a  flat  in  a  sharp  key,  and  as  a  sharp  in  a   fl.it  key. 


GOODRICH  S    ANALYTICAL    HARMONY.  ig 

In  the  first  measure  d  and  a-flat  constitute  an  imperfect  5th ;  in 
the  second,  d  and  g-sharp  form  an  augmented  4th.  The  student 
should  supply  the  theoretical  names  of  the  following  ascending 
and  descending  intervals : 


7th.  3d. 


Nor. 


Ex.  23. 


A  thorough  knowledge  of  scales  will  enable  one  to  name  all  of 
these  intervals  correctly.  For  instance,  take  g  and  e-flat.  In 
G-major  e  is  natural,  and  therefore  a  major  6th.  In  G-minor  e 
\s>flat,  and  is  a  minor  6th  above  g. 

Some  of  the  names  generally  applied  to  intervals  appear  incon- 
sistent; the  author  has,  therefore,  changed  them.  However,  this  is 
a  matter  of  individual  opinion,  and  does  not  affect  the  intervals. 


Chapter  III. 


FORMATION  OF  MAJOR  CONCORDS. 


THE  most  euphonious  intervals  in  music  are  major  and  minor 
thirds  and   their  inversions,  minor  and  major  sixths.      These 
may  succeed  each  other  without  involving  false  progressions,  though 
if  they  be  too  long  continued  the  ear  becomes  satiated  with  their 
consonant  effect. 

Inasmuch  as  the  normal  scales  are  composed  of  whole  and  half 
steps,  the  thirds  are  naturally  large  and  small.  Hence  a  phrase  like 
the  following  is  perfectly  euphonious,  because  the  ear  recognizes  all 
the  sounds  as  belonging  to  the  major  scale  of  F: 


Ex.  24. 


2o  GOODRICH'S  ANALYTICAL  .-«.KMONY. 

The  major  and  minor  thirds  here  succeed  one  another  so  natu- 
rally that  none  but  a  cultivated  ear  can  recognize  a  difference ;  yet  a 
major  $d  is  one  chromatic  step  larger  than  a  minor  3d,  and  the 
difference  between  the  major  and  the  minor  mode  is  very  pro- 
nounced. 

This  somewhat  superficial  inquiry  into  the  character  of  large  and 
small  thirds,  and  the  promiscuous  order  in  which  they  naturally 
occur  in  a  scale,  is  intended  to  form  the  basis  of  an  elementary 
knowledge  of  concords. 

The  natural  origin  of  what  we  call  Harmony  may  be  ascribed  to 
the  philosophy  of  sound.  Acoustics  has  revealed  many  curious 
and  interesting  phenomena  of  a  musical  character,  and  the  student 
should  possess  at  least  some  slight  knowledge  of  the  general 
results  attained  through  purely  scientific  investigation. 

As  color  is  an  inherent  property  of  light,  so  is  harmony  an 
.inherent  property  of  sound. 

Aside  from  the  researches  of  philosophers  and  physicists,  musical 
theorists  have  for  centuries  past  demonstrated  the  fact  that  nearly 
all  musical  sounds  are  composite.  In  other  words,  the  single  funda- 
mental tone  of  a  bell,  string  or  tube  generates  other  tones  related  to 
the  fundamental  by  natural  laws.  Several  of  these  overtones  were 
known  to  the  Greek  philosopher  and  theorist  Pythagoras,  who 
evolved  from  them  something  of  a  system ;  in  fact  the  Egyptians, 
before  the  existence  of  the  Greek  nation,  possessed  a  scale  similar 
to  our  normal  major  scale.  As  the  ratios  of  this  old  scale  are  more 
nearly  perfect  than  our  tempered  scale,  it  is  evident  that  the  former 
was  developed  according  to  mathematical  deduction. 

The  most  important  series  of  harmonics  or  partial  tones  are 
those  of  the  natural  horn,  produced  by  variable  quantities  of  wind- 
pressure,  without  the  aid  of  artificial  valves.  These  tones  are  num- 
bered consecutively  from  i  to  6,  in  the  order  of  their  arrangement : 


Ex.  25. 


&          a         3     •>      4         5         6 


No.  I  Is  the  fundamental,  the  others  are  inherent  elements  or 
effects  of  this  generator.  There  are  other  partial  tones,  though 
unrecognizable  except  with  the  aid  of  a  resonator.  The  above  har- 
monics can,  however,  be  heard  under  favorable  circumstances  as 
overtones,  and  on  the  horn  they  all  come  out  distinctly  by  means 
of  variable  wind  and  lip  pressure. 


GOODRICH 'S    ANALYTICAL    HARMONY. 


21 


The  vibration  numbers  of  these  harmonics  are  connected  by  a 
simple  law  of  acoustics.  If  the  lowest  tone  makes  100  vibrations  a 
second,  (2)  will  make  200,  and  so  on,  in  the  same  space  of  time. 
"  The  proportion  remains  the  same  whatever  the  fundamental  may 
be,  and  thus  it  is  plain  that  the  above  harmonics  belong  to  a  fixed 
series  of  overtones." — Taylor. 

This  is  why  such  tones  as  the  following  are  so  easily  sounded 
upon  the  tubular  instruments,  like  the  horn,  trumpet,  trombone, 
and  cornet : 


Ex.  26. 


-ay 


The  first  harmonic  of  our  series  (2)  is  naturally  the  most  impor- 
tant, as  its  vibration  number  corresponds  to  i,  according  to  the 
equal  ratios  y,  f ,  f ,  and  so  on.  In  plainer  terms,  the  octave  makes 
two  vibrations  to  every  one  of  the  fundamental.  Hence  the 
Greeks,  who  did  not  employ  harmony  as  we  understand  it,  sang  in 
unison ;  and  as  there  is  a  difference,  expressed  by  the  ratio  i  :  2, 
between  the  voices  of  men  and  women,  the  result  was  octave  pro- 
gressions. 

The  next  most  important  interval  of  the  series  (Ex.  25)  is  the 
fifth,  No.  3.  The  ratio  is  2  :  3,  or  f .  This  was  the  interval  next 
employed  for  simultaneous  progressions  about  the  year  800,  A.  D. 
A  brief  specimen  of  this  diaphony  is  presented : 


Ex.  27. 


Yt  consists  of  a  series  of  octaves  with  the  fifths  added  above  the 
iundamental  base.  The  example  sounds  crude,  almost  barbarous  to 
modern  ears,  and  it  is  one  of  the  numerous  failures  attending  the 
efforts  of  theorists  to  establish  a  plastic  art  upon  an  abstract  founda- 
vion,  to  the  sacrifice  of  a  higher  law  than  mathematical  deduction. 

The  next  progress  was  in  the  direction  of  counterpoint,  which 
became  highly  developed  before  harmony  was  known  in  distinct 
chord  formations.  Major  and  minor  thirds  having  appeared  in  the 
vocal  part-music  of  the  i5th  and  i6th  centuries,  composers  such  as 
Tallis  and  Viadana  had  but  to  include  the  thirds  with  the  fifths  in 
order  to  produce  major  and  minor  triads.  These  chords  wiU  firs 
engage  the  student's  attention. 


22  GOODRICH'S  ANALYTICAL  HARMONY. 

By  referring  to  Ex.  25  it  will  be  seen  that  the  harmonics  form  a 
perfect  major  chord. 

Select  i,  3,  5  from  the  series,  omitting  the  duplicates: 


Ex.  28. 


3-*>- 


This  is  the  major  chord  of  F,  in  open  position 


Arrange  it  in  close  position  for  convenience  and  simplicity : 

29'  Eg&JLlAg The  same  numbers  apply  in  both  instances, 


for  here  i  is  the  root,  3  is  the  third,  and  5  is  the  fifth.  The  funda- 
mental (or  generator)  is  the  root,  the  tone  upon  which  the  chord  is 
founded.  As  the  chord  originated  in  this  way,  and  as  the  root  gives 
to  the  combination  its  name,  the  author  treats  this  as  its  first  posi- 
tion. An  analysis  shows  that  this  chord  contains  a  major  3d  (/  to 
a)  and  a  normal  5th  (/  to  c).  This  is  fundamental  enumeration. 
Componently  the  chord  contains  a  major  and  a  minor  3d.  The 
results  are  the  same.  This  is  known  as  a  major  concord,  or  as  the 
consonant  triad  of  F  -major.  It  is  the  most  important  chord,  espe- 
cially when  founded  upon  the  tonic,  as  in  the  last  example. 

As  this  system  is  based  upon  the  artistic  results  of  actual  compo- 
sition, the  author  believes  all  arguments  futile  that  attempt  to  dis- 
credit these  results,  or  that  call  in  question  the  means  employed  by 
inspired  composers.  We  not  only  accept  these  results,  but  find 
them  excellent. 

There  are  but  two  other  major  concords  to  be  found  within  the 
limits  of  a  scale.  The  second  is  founded  upon  the  fourth  of  the 
scale,  and  is  known  as  the  Sub-dominant  harmony  : 

It  contains  the  same  theoretical  intervals  as  the 


tonic  chord,  a  major  3d  and  a  normal  5th.     The  root  is  B-flat,  and 

the  5th  is  in  unison  with  the  key-tone  F.     Therefore  there  is  an  ini- 

• 

portant  connecting  link  between  them,  thus  : 

This  is  true  in  whatever  situation  the  chords  may  appear. 

The  third  major  chord  is  based  upon  the  fifth  degree  of  the 
scale.     It  is  known  as  the  Dominant  harmony,  and  is  constructed  in 


GOODRICH'S  ANALYTICAL  HARMONY. 


^~:~       Counting    from 


the  same  manner  as  the  others  :    Ex.  32. 


the  root  C,  it  contains  a  major  3d  and  a  normal  5th.     The  root  of 
this  chord  is  in  unison  with  the  5th  of  the  tonic  chord,  and  the  two 


are  thus  connected  :    EX.  33. 


iscE:!    Some  modern  theorists 


present  these  three  fundamental  harmonies  in  this  manner: 


Ex.34. 


rfcr     — ^_ 


The  ulterior  design  of  this  arrangement  is  to  illustrate  the  hypothe- 
sis of  an  under-scale  system,  but  as  the  chords  can  not  succeed  one 
another  in  this  manner,  it  is  not  adopted  here.  The  regular  mode 
of  progression  for  these  harmonies  is  here  given: 


Ex.  35. 


Every  tone  in  the  scale  is  embraced  in  these  three  chords,  and 
countless  numbers  of  popular  pieces  contain  no  other  harmonies. 

Write   the  tonic,  sub-dominant,  and  dominant  chords  in  their 
original  positions  in  at  least  six  other  scales. 


Chapter  IV. 


FORMATION  OF  MINOR  CONCORDS. 


minor  concord  is  generally  considered  as  a  derived  har- 
•*-  mony.  Certain  writers  argue  that  it  has  no  place  among 
fundamental  harmonies,  and  much  discussion  has  taken  place  re- 
specting its  origin  and  character.  But  it  would  be  unprofitable  to 
enter  into  the  controversy  here.  The  minor  triad  has  been  freely 
used  by  all  the  great  composers,  and  the  only  important  problem  is 
to  ascertain  how  it  has  been  treated. 


24         GOODRICH'S  ANALYTICAL  HARMONY. 

The  major  scale  presents  sufficient  material  for  the  construction 
of  minor  chords. 

Though  the  intervals  are  all  normal  or  perfect  (counting  up- 
ward from  the  fundamental),  the  internal  arrangement  yields  more 
minor  than  major  thirds  : 


mi. 


mi.      ma.      ma.      mi. 


EX. 


Here  are  four  minor,  and  but  three  major  thirds,  for  the  reason  that 
the  seventh  triad  contains  two  minor  thirds,  b  to  d,  and  d  to  /.* 
If  a  fifth  be  added  to  the  first  six  duophonic  chords  there  will  result 
six  concords.  Three  of  these  will  be  major  and  three  minor.  It 
must  be  understood  that  a  normal  5th  contains  a  major  and  a  minor 
3d  (or  vice  versa),  not  two  major  thirds;  for  this  would  result  in  a 


discord :    Ex  37- 


There  are,  therefore,  three  minor  chords 


in  every  major  scale,  founded  upon  the  second,  third,  and  sixth 
degrees,  with  a  3d  and  a  5th  added  above.  Write  these  in  their 
original  positions.  *  *  * 

The  relative  minor  of  any  major  is  situated  a  small  3d  below 
the  latter,  and  the  minor  chord  will  contain  a  minor  and  a  major 
3d,  instead  of  vice  versa.  The  connections  between  the  chords 
appear  to  better  advantage  in  this  form: 


Ex.  38. 


F- 


The  principal  points  to  be  observed  are  : 

1.  That  the  relative  minors  are  in  each  instance  located  a  small 
3d  below  the  relative  majors,  as  shown  by  the  base. 

2.  That  two  notes  of  each  major  chord  occur  in  the  relative 
minor  chord.     These  are  notes  of  connection,  as  will  appear  here- 
after.    By  combining  the  concords  which   have  been  discovered, 
there  will  be  six :  three  major  and  three  minor.     These  are  to  be 


*The  preponderance  of  minor  thirds  in  tin's  example  is  all  the  more  remarkable,  con- 
sidering that  the  major  scale  of  one  octave  contains  five  major,  and  but  two  minor  seconds. 


GOODRICH'S  ANALYTICAL, HARMONY. 


written  with  the  understanding  that  the  3d  and  5th  are  to  be  added 
above  each  of  the  first  six  degrees  of  the  major  scale.  *  *  * 

It  is  the  author's  purpose,  during  the  first  twenty  chapters,  to 
employ  concords  only,  and  as  these  must  contain  a  normal  5th  and 
a  major  or  a  minor  3d,  it  is  evident  that  but  two  species  of  concords 
are  recognized. 

The  simplest  manner  of  determining  the  character  of  these  con- 
cords (whether  major  or  minor)  is  to  remember  that  i,  3,  5  in  every 
major  scale  constitute  a  major  concord,  and  that  i,  3,  5  in  every 
minor  scale  form  a  minor  concord.  The  next  example  shows  all  the 
consonant  triads  in  F-major  : 


mi.      mi. 


Ex.  39. 


ma.      mi. 


(The  student's  exercise  should  correspond  to  this.)  Each  chord 
appears  in  its  first  position,  with  the  root  at  the  bottom. 

This  merely  shows  the  original  formation  of  the  various  triads, 
for  they  can  not  follow  one  another  in  this  manner. 

The  triad  founded  upon  the  "  leading-tone  "  does  not  contain  a 
normal  5th,  and  therefore  can  not  be  a  concord  : 


Ex.  40. 


The  interval  from  e  to  b-flat  is  what  the  author 


terms  an  imperfect  5th  (generally  called  diminished),  and  as  all  theo- 
rists agree  that  it  is  not  a  concord  it  is  excluded  from  this  part  of 
the  work. 

The  names  of  the  minor  harmonies  are  frequently  called  after  the 
degrees  of  the  scale  upon  which  they  are  based.  These  technical 
terms  in  the  major  scale  are  : 

Super-tonic  (2),  the  next  above  the  tonic; 

Mediant  (3),  midway  between  Tonic  and  Dominant; 

Sub-mediant  (6),  the  same  distance  below  the  tonic  that  the 
mediant  is  above  it. 

Sometimes  they  are  called  by  terms  of  relation,  as  relative 
minor  of  the  tonic  (No.  6  in  Ex.  39),  relative  minor  of  the  sub- 
dominant  (No.  2),  and  relative  minor  of  the  dominant  (No.  3). 

As  relative  major  and  relative  minor  scales  have  the  same  signa- 
ture it  is  often  convenient  to  mention  them  in  this  manner. 

Examples  similar  to  No.  39  should  be  written  in  all  the  major 
scales  and  named  according  to  their  major  or  minor  character. 


GOODRICH'S  ANALYTICAL  HARMONY. 


Chapter  V. 


MAJOR  AND  MINOR  CONCORDS   RE-ARRANGED. 

IN  the  previous  chapters  each  concord  has  appeared  in  its  first,  or 
original  position.     But  as  every  chord  has  as  many  positions  as 
tones,  or  letters,  the   order  of  the  intervals  may  now  be  changed. 
This  can  be  illustrated  more  readily  by  re-arranging  the  letters 
which  apply  to  the  three  tones  of  a  concord. 

Beginning  with  the  first  position  of  the  C  chord,  the  letters  will 


read  from  the  lowest,  C,  e,  g:     Ex-  4I-Nfrr— ^r -       Next  begin  with 


the  second  letter  (e)  placing  C  last :  e,  g,  C.     This  is  the  second  posi- 

_n ? 

tion  of  the  C  chord ;  in  notation  thus :     Ex-  42-  F/jfe-^—       As  no 


new  letters  have  been  added  it  is  still  the  C  chord,  with  the  root 
placed  above  instead  of  below. 

By  placing  the  3d  uppermost  the  third  position,  g,  C,  e,  is  ob- 


tained:     Ex-  43-  pyfo—  Each  of  the  six  triads  is  to  be  re-ar- 

l~  V:  I/       ^  __  _"~l 

ranged  in  the  same  manner,  in  regular  order.     Elementary  students 
are  advised  to  re-arrange  the  chords  first  by  means  of  letters  in  this 

{D  f  a,  i. 
f  a  D,  2.     The  figure  shows  the  position  and  the  capital 
a  D  f,  3. 
indicates  the  root. 

Other  combinations  are  possible  with  the  three  letters  of  a  triad, 
but  these  involve  open  positions,  or  "  dispersed  harmonies,"  which 
are  not  to  be  employed  at  present.  These  are  produced  by  inverting 
the  middle  interval  of  each  close  position  : 


Ex.  44. 

A,  c,  and  e  are  close  positions ;  b,  d,  and  f  are  open  positions.*    These 

*In  addition  to  these  the  author  uses  what  he  terms  half-open  positions.    Dispersed 
harmony  is  reserved  for  Harmonic  Counterpoint. 


GOODRICH'S  ANALYTICAL  HARMONY. 


27 


are  mentioned  here  to  show  the  open  positions  and  to  caution  against 
their  present  employment.  The  word  Inversion  is  not  used  for 
these  re-ananged  chords,  but  is  reserved  for  a  future  chapter,  where 
the  base  L'.as  another  tone  than  the  root. 

In  re-arranging  the  six  triads  always  begin  with  the  first  posi- 
tion, leading  the  letters  upward  as  in  chord  formation.  Add  no  new 
letters  to  the  three  which  represent  a  certain  concord,  and  follow 
these  letters  in  their  regular  order,  avoiding  open  positions.  Num- 
ber the  positions  in  each  re-arrangement.  After  completing  the 
exercise  by  means  of  letters,  a  corresponding  example  must  be 
written  in  notation.  Do  not  forget  that  the  root  of  a  chord  remains 
the  same  during  the  different  re-arrangements,  and  until  some  new 
element  appears.  A  simple  example  will  illustrate  this  fact : 


G  tnaj. 


E  mi. 


Ex.  45- 


o-  ' 
Root. 


In  the  first  measure  the  harmony  of  G-major  prevails,  and  G  is  the 
natural  foundation  of  these  tones.  In  the  second  measure  a  new 
element  is  introduced,  which  results  in  the  chord  of  E-minor,  The 
upper  parts  progress  through  the  different  positions  of  this  chord, 
and  E  remains  in  the  base  as  root  and  natural  foundation  of  the 
upper  harmony.  The  six  concords  in  each  major  scale  should  be 
re-arranged  in  three  close  positions,  as  already  explained.  After  two 
or  three  re  arrangements  have  been  made  with  letters,  they  can  be 
omitted  and  the  remainder  written  in  notation  only. 
A  complete  example  is  given  for  comparison  : 


Ex    |6. 


E-flat.  F-mi.  G-mi.  A-flat.  B-flat. 

„    g.         *^-,^    ^^.^g^ 


C-mi. 


W     '  g    g          l%-~- i-^1— 

•^          -  I      2      3       1      2      3       1      1 


23        123        123         123 


28  GOODRICH'S  ANALYTICAL  HARMONY. 


Chapter  VI. 


HARMONIZATION  IN  THREE  PARTS. 

HARMONIC   Progression,  or   Chord   Succession,  is  the   act   of 
moving   from  one  tone-combination  to  another,  according   to 
certain  principles.     What  chords  may  follow  one  another  is  not  so 
important  a  question  as  how  they  shall  follow. 

It  has  been  observed  that  from  the  natural  tones  of  the  major 
scale  three  major  and  three  minor  concords  can  be  formed,  each 
of  which  has  three  close  positions  resulting  from  re-arrangement. 

As  a  preliminary  solution  of  the  problem  of  Harmonization  the 
author  has  devised  the  following  method. 

The  first  object  is  to  inquire  how  many  chords  may  be  written 
beneath  each  note  in  the  major  scale.  If  the  key  of  G-major  is 
selected,  begin  with  the  key-tone,  i. 

The  student  should  have  for  reference  a  chart  containing  the  six 
triads  in  G,  re-arranged  in  three  close  positions,  which  must  be  con- 
sulted in  order  to  answer  the  necessary  questions,  thus : 
_£_«.«•        -      -*•        •     B 

Ex.  47- 


1.  Write  a  treble  clef  with  the  signature  of  G,  and  mark  g  on 
the  second  line  for  illustration.     The  number  of  chords  that  are  to 
accompany  this  fixed  tone  depends  upon  how  many  of  the  six  con- 
cords contain  g.     (It  will  be  sufficient  to  refer  to  the  first  position 
of  each  triad,  as  the  re-arrangement  in  each  measure  represents  the 
same  chord.) 

2.  How  many  chords  among  the  six  in  this  key  contain  £•?* 

3.  Write  g  as  many  times  as  there  are  chords  containing  g, 
These  should  be  tied  together,  representing  a  soprano  part,  which 


r--   __ 

remains  uppermost  throughout,  thus  :      Ex-  48«-  Fifo      -^^ijF^^F 

Lt2_ 

The  three  triads  containing  g  are  to  be  written  beneath  this  fixed 
note,  g  forming  a  part  of  each  chord. 

*  Whether  g  occurs  above  or  below  matters  not  ;  it  islsufficient  that  g  occurs  in  a  certain 
chord. 


GOODRICH'S  ANALYTICAL  HARMONY.  29 

4.  What  are  the  three  chords  containing  g  ? 

[Mention  them  in  regular  order  by  their  root  names,  specifying 
whether  they  are  major  or  minor.] 

5.  What  is  the  first  chord  containing  gt 

6.  What   position  of  this  chord  has  g  uppermost?     (See  first 
measure  of  chart.) 

7.  The  root-note  already  appears  in  the  soprano,  therefore  add 
the  other  two  notes  immediately  beneath  the  stationary  note. 

8.  What  is  the  second  chord  containing  g  ? 

9.  What  position  according  to  the  chart  has  g  uppermost? 

10.  Write  the  remainder  of  the  chord  beneath  the  second  tied 
note. 

11.  What  is  the  third  chord  containing  g? 

12.  What  position  shall  be  used  in  order  to  have  g  at  the  top? 

13.  Write  this  beneath  the  third  tied  note. 

After  completing  this  illustration  of  the  note  g,  draw  a  double 
bar,  and  add  capital  letters  below,  representing  the  root  of  each 
chord.  *  *  *  Supposing  this  much  to  have  been  accomplished, 
a  complete  example  is  presented  for  comparison : 


Ex.  48-5. 


The  numbers  included  above  the  chords  show  the  position  of  each. 
These  progressions  are  perfectly  correct. 

The  following  harmonic  progressions  have  been  made :  G  to  C 
and  from  C  to  E-minor.  G,  appearing  above  as  part  of  each  chord, 
serves  as  connecting  link  in  this  chain  of  harmonies,  and  at  the 
same  time  shows  that  the  tonic  of  G-major  may  be  accompanied 
with  any  or  all  of  these  chords.  The  first  chord  appears  in  its 
second  position,  the  "second  chord  appears  in  its  first  position,  and 
the  last  chord  is  in  its  third  position. 

As  a  further  result,  observe  that  the  stationary  upper  note  is 
root  of  the  first  chord,  5th  of  the  second,  and  3d  of  the  last  chord. 

Compare  these  observations  with  Ex.  48^  until  they  are  thor- 
oughly comprehended. 

Another  circumstance  to  be  noted  is,  that  no  part  moves  more 
tnan  a  major  or  a  minor  2d  up  or  down. 

The  second  note  of  the  scale  is  now  selected  for  illustration. 

*  The  last  chord  appears  an  octave  lower  than  in  the  chart,  but  the  two  are  to  be  consid- 
ered identical. 


3O  GOODRICH'S  ANALYTICAL,  HARMONY. 

How  many  chords  in  this  key  contain  a  ? 

\Vrite  second  space  a  as  many  times  as  there  are  concords  t  on- 
taining  that  note. 

Mention  these  chords  in  regular  order,  and  specify  whether 
major  or  minor. 

What  is  the  first  chord  containing  a  ? 

What  position  is  required  when  a  is  uppermost.  (See  chnrt.) 
Write  this  chord  beneath  a.  What  is  the  second  chord  containing 
#  ?  What  position  is  required  in  order  to  retain  a  at  the  top? 

Write  this,  including  the  numbers  above,  and  the  capital  letters 
below  to  indicate  the  roots.  In  harmonizing  the  third  note  of  the 
scale  the  formula  of  questions  and  the  means  of  answering  remain 
the  same. 

The  tied  note  should  be  written  as  many  times  as  there  are 
chords  in  which  it  occurs.  This  fixed  tone  is  always  to  be  con- 
sidered as  soprano  or  upper  voice,  and  the  chords  written  beneath 
it  must  correspond  to  the  chart.  In  each  instance  include  the  cap- 
ital letters  to  show  the  roots,  and  number  the  position  of  each  chord 
as  indicated  in  the  chart. 

All  this  tends  to  give  the  student  a  thorough  acquaintanceship 
with  the  positions  and  character  of  concords,  and  the  author  ad- 
vises that  no  detail  be  neglected. 

The  fourth,  fifth,  sixth,  and  seventh  notes  of  the  scale  are  to  be 
illustrated  in  the  same  manner.  *  *  * 

On  account  of  the  imperfect  triad  being  here  excluded,  it  will 
appear  that  the  2d,  4th,  and  yth  degrees  of  the  scale  admit  but  two 
chords  each  in  their  illustration  ;  whereas  all  the  other  degrees  may 
be  accompanied  with  three  chords.  The  yth  tone  may  be  harmon- 
ized with  two  chords,  though  no  consonant  triad  is  founded  upon 
that  tone. 

Every  tone  in  the  following  major  scales  Should  be  treated  in 
the  same  manner :  A,  B,  F,  E-flat. 

The  last  two  keys  might  be  attempted  v.-ithout  the  aid  of  a 
chart,  provided  the  student  is  thoroughly  familiar  with  all  the  con- 
cords in  their  various  re-arrangements. 


For  the  benefit  of  those  who  have  not  the  aid  of  a  teacher  ^ 
completed  example  is  appended,  in  order  that  the  pupil's  work  may 
be  compared  with  the  printed  example : 


S  ANALYTICAL   HARMONY. 


Ex.  49- 


La  Si 

1  ~*  ~$    .  U     ^ 

L±g=g=r-^=g: 

; *^ ^< ^ »^- 


The  syllables  above  are  included  to  show  what  notes  of  the 
scale  are  illustrated,  and  ia  \vhat  mannei.  This  will  serve  a  special 
purpose  hereafter. 

This  example  is  not  continuous.,  each  measure  being  considered 
separately. 


GOODRICH  S   ANALYTICAL   HARAIGtfV. 


PART  II 


Chapter  VII. 


THEORY  OF  HARMONIC  PROGRESSION. 

CHORD  SUCCESSIONS  RE-ARRANGED. 

A  SOMEWHAT  superficial  knowledge  of  chord-progressions 
£*•  having  been  acquired,  a  more  general  and  thorough  system 
will  now  be  introduced. 

1.  Any  note  occurring  in  two  different  chords  is  called  a  con- 
necting note. 

2.  Every  connecting  note  i<    to  be  tied,  or  sung  by  the  same 
voice-part  in  both  chords. 

3.  When  there  are  two  connecting  notes  between  two  chords 
in  progression,  both  notes  are  to  be  tied,  or  rer~ain  stationary : 

*= 


Ex.  sort. 


I 


G  is  the  connecting  link,  being  common 


to  the  three  chords.  This  note  being  in  the  soprano  part,  the  C 
chord  must  appear  in  its  first  position  in  order  to  retain  the  note  of 
connection  in  the  upper  part. 

Between  the  C  and  the  E  chords  there  are  two  connecting  notes, 
e  and  g.     As  a  rule  these  should  be  written  first  : 


Ex>  5°'  — --      The  note  wanting  in  the  last  chord  is  its 


fifth,  therefore  the  contralto  part  descends  a  minor  2cl,  from  C  to  bi 

F  V      C  T     I — M ^^ 1 

pfe    %^jg~~l    Only  close  positions  are  to  be  used  at  present, 
and  the  voice-parts  must  not  progress  up  or  down  more  than  a  2d. 


GOODRICH'S  ANALYTICAL  HARMONY. 


33 


The  first  measure  from  Ex.  49  is  selected  for  re-arrangement 
in  two  other  positions.  The  next  note  of  the  G  chord  above  is  b, 
which  will  become  the  soprano-part,  the  chord  reading  d,  g,  b.  In 
this  position  the  same  progressions  should  be  made  as  in  Ex.  50^, 
i.  e.,  from  G  to  C  and  C  to  E.  It  would  be  well  to  use  three  treble 
staffs,  one  above  the  other.  The  arrangement  of  the  lowest  should 
be  the  same  as  Ex.  49.  A  brief  indication  of  the  design  of  the  first 
few  measures  is  here  given : 

3.        GCEADGBE 


Ex.  52. 


•ytt  g    g 

^ 

^^_^^ 

fn  —  ^  —  -  —  ~s~ 

~S  ^  — 

-  ^*  ^^^--^  ^  

V-\J        &        &        & 

•^••^^ 

jr       —  -  •  —  •- 

2.        GCEADGBE 

W      0* 

_.             ^^ 

JF    •        ^»o         ^^ 

S^ 

yl^-    *j>^  —  .^. 

mj^^             fS^*  *•*•*  f^>-^*^*^-  j*^? 

&            & 

c^T            c^             &/ 

v-  ly        ~*r       5p[       *^ 

1.        GCEADGBE 

n  i 

V  it 

^  »^.     ,—  ^ 

_  —  •  _ 

_.,        j^?        j<? 

CIEV       .  ,-.X3     X3  ^5  — 

^            ^ 

J^                _n-            J<? 

etc. 


•»••*•    -BT  •*•  - - 

The  progressions  at  2  and  3  being  the  same  fundamentally  as  at  I, 
are  to  be  accomplished  in  the  same  manner;  that  is,  2  and  3  are 
re-arrangements  of  i. 

The  stationary  note  g,  which  was  first  in  the  soprano,  appears 
throughout  No.  2  in  the  mezzo-soprano  part.  Consequently  the 
middle  voice  continues  to  sing  g,  b  ascends  a  half  step  to  c,  while 

2. 

.— 6-S- 

d  in  the  contralto  ascends  a  w?hole  step  to  e :   Ex.  53. 


The  next  progression  is  from  C  to  E.  By  consulting  No.  i 
it  will  be  seen  that  there  are  two  connecting  notes,  e  and  g.  In 
No.  2  these  appear  in  the  contralto  and  mezzo-soprano.  Therefore 
while  those  notes  remain  stationary  the  soprano  descends  a  half 
step  from  c  to  b,  in  order  to  complete  the  E-minor  chord : 


Ex.  54. 


These  progressions  should  now  be  written 


in  the  upper  staff  (No.  3).  The  stationary  note,  g,  which  first 
appeared  in  the  soprano,  and  then  in  the  mezzo-soprano,  now 
appears  throughout  in  the  contralto  part.  Therefore  the  three 
chords  are  to  be  written  above  this  note. 


34 


GOODRICH'S  ANALYTICAL  HARMONY. 


Always  write  the  connecting  notes  first,  and  move  the  other  parts 
as  little  as  possible,  without  skipping.  No.  3  is  accomplished  in 
the  same  manner  as  was  No.  2. 

The  second  measure  comes  next,  and  is  to  be  treated  in  the  same 
manner.  The  note  of  connection  occurs  at  i  in  the  upper  part,  at 
2  in  the  middle  part,  and  at  3  in  the  lower  part.  By  writing  the 
connecting  note  first  and  tying  it,  the  remainder  of  the  chord  is 
easily  filled  in,  because  the  connecting  note  fixes  the  position  of  each 
chord. 

Each  measure  of  Ex.  49  is  to  be  re-arranged  in  two  other  posi- 
tions, as  indicated.  2  and  3  are  duplicates  of  i,  being  the  same 
harmonic  progressions  in  different  positions. 

It  is  well  at  first  to  include  the  same  capital  letters  in  each 
corresponding  measure  in  order  to  show  that  the  chords  are  the 
same.  (See  Ex.  52.) 

Elementary  harmony  students  are  so  inclined  to  misapply  the 
principles  of  connecting  notes  that  explanation  on  this  point  can 
not  be  too  explicit.  The  following  progressions  show  the  common 


mistake : 


Ex.  55. 


Notwithstanding  the  ties,  the  first 


c     G     D 

note  in  common,  g,  does  not  remain  in  the  same  voice-part,  but 
skips  up  to  b.  In  the  next  progression  d  is  the  connecting  link ; 
but  it  goes  to  /,  instead  of  remaining  stationary.  To  prove  the 
incorrectness  of  these  progressions  they  will  be  written  in  score: 

Soprano. 


Ex.  56. 


Mezzo-Soprano. 


tgl    ^ 


COD 
Contralto. 


v- 


I 


-&-—&- 


I 


(a)  is  taken  directly  from  the  last  exercise,  and  is  here  forbidden. 
At  (b)  the  progressions  are  correct,  since  the  connecting  notes 
remain  in  the  same  voice-parts.  In  a  single  staff  this  would  appear  as 


toilows:     Ex-57. 


Compare  this  with  the, 


GOODRICH'S  ANALYTICAL  HARMONY.  35 

pie  in  score.  The  rule  for  connecting  notes  is  often  set  aside  by 
composers,  and  the  exceptions  will  be  duly  explained  in  their 
proper  place.  It  will  be  well  to  remember  until  the  restriction  in 
removed  that  no  two  chords  should  occur  successively  in  the  same 
position.  The  last  measure  of  the  re-arrangements  is  presented  for 
comparison : 


The  connecting  notes  being  written  first  and  tied,  nothing  remains 
but  to  move  b  down  to  a  in  each  arrangement. 

Two  or  three  arrangements  in  other  keys  should  be  made. 


Chapter  VIII. 


HARMONIC  PROGRESSION  IN  FOUR  PARTS. 

ADDITION   OF  THE  FUNDAMENTAL  BASE. 

BY  adding  a  base  to  the  previous  three-part  progressions,  four- 
part  harmony  will  result. 

It  is  both  natural  and  proper  to  suppose  that  as  the  base  is  the 
foundation  of  harmony  it  shall  consist  of  fundamental,  or  root, 
tones.  For  the  present  this  arrangement  wiQ  be  followed,  without 
regard  to  the  position  of  the  chord.  In  order  to  determine  the  root 
of  a  chord  arrange  it  all  upon  lines,  or  all  in  spaces,  in  which  case 
the  root  will  be  below;  for  root  signifies  foundation,  or  generator. 

Some  experience  has  already  been  acquired  in  this  matter ;  but 
in  addition  to  this  the  author  advises  all  music  students  to  become 
accustomed  to  the  appearance  and  the  sound  of  chords  in  their  dif- 
ferent positions.  Chords  in  their  first  position  consist  of  two  thirds : 

Ex.  59. 


GOODRICH 'S    ANALYTICAL    HARMONY. 


The  intervals  of  the  second  position  may  be  described  as  a  3d  and  a 
4th: 


Ex.  60.  : 


The  third  position  consists  of  a  4th  and  a  3d : 


Ex.  61. 


m 


Both  the  eye  and  the  ear  should   be  trained  to  distinguish  these 
positions  on  the  instant. 

The  roots  of  these  last  three  examples  are  the  same,  to  wit,  G, 
A,  B,  C.     One  more  preliminary  example  will  suffice : 


Ex.  62. 


Root. 


The  tone  upon  which  this  chord  was  founded  and  which  gives  it  its 
name  is  G.  Therefore,  while  the  treble  parts  pass  through  the 
different  positions  of  the  chord,  the  base  remains  on  the  foundation 
tone,  the  root. 

The  student  may  now  write,  in  the  base  staff,  the  roots  of  the 
following  chords : 

Ex.  63. 


A  few  instructions  are  necessary  with  regard   to  the   management 
of  the  base : 

1.  It  is  to  be  given  the  root  of  each  chord. 

2.  It  should  not  skip  up  or  down  more  than  a  $th. 

3.  It  must  not  appear  above  an}'  of  the  other  parts. 

PROGRESSIONS    REVERSED. 

Taking  as  a  foundation  the  previous  progressions,  additional 
chord  combinations  will  now  be  given.  Begin  in  A-major,  and 
write  as  many  a?s  as  there  are  chords  containing  a.  This  exercise 
will  require  two  staffs,  treble  and  base.  The  first  chord  containing 
a  is  to  be  written  with  that  note  uppermost.  This  a  is  to  be  a 


GOODRICH  S   ANALYTICAL    HARMONY. 


37 


fixed  tone  during  the  first  example.    D  is  the  next  chord  containing 
a.     As  the  soprano  is    tied,  the  D  chord  must  appear  with  a  at  the 


top:  Ex 


The  third   chord   is  F%   minor. 


There  are  two  notes  in  common.  Retaining  these  in  the  soprano 
and  mezzo-soprano,  the  contralto  descends  to  c%  and  the  chord  is 
complete,  for  each  chord  must  here  appear  with  a  at  the  top".  The 
chords  employed  are  A,  D,  and  F-sharp  minor*  In  the  second 
combination  the  order  is  changed  to  A,  F-sharp,  and  D,  with  two 

connecting  tones   between  the  chords :  Ex*  65- 

The  third  is  D,  F-sharp,  and  A ;  the  fourth  D,  A,  F-sharp.  Two 
more  combinations  are  possible,  beginning  each  time  with  the  minor 
chord  :  F-sharp,  A,  D,  5  ;  F-sharp,  D,  A,  6.  These  are  to  be  written 
in  accordance  with  the  connecting  tone  principle,  keeping  a  upper- 
most throughout. 

The  base  is  now  to  be  written.  Simply  ascertain  the  root  and 
write  that  note  beneath  each  chord  which  it  represents.  Two 
measures  are  presented  for  comparison  : 


Ex.  66. 


~&L 


Roots. 


(Where  there  are  duplicated  bases,  either  one  is  correct.) 

The  next  exercise  consists  in  illustrating  the  second  tone  of  the 
scale.  There  are  but  too  chords  available  for  this  purpose.  Write 
these  with  the  connecting  note  (b)  uppermost.  Then  reverse  the 
order,  B  -minor  to  E;  E  to  B  -minor. 

The  third  tone  of  the  scale  is  next  in  order  and  admits  six  com- 
binations, according  to  the  principles  already  explained. 

After  the  chords  are  completed,  the  roots  are  to  be  added  in  the 
base,  making  four-part  harmony. 

In  the  same  manner  every  tone  in  the  A-major  scale  is  to  be 
illustrated  and  combined. 


*  The  major  chord,  being  more  natural  than  the  minor,  is  presupposed  when  only  the 
root  name  is  given.    But  when  a  minor  chord  is  intended  it  must  be  so  expressed. 


38  GOODRICH'S  ANALYTICAL  HARMONY. 

Where  a  certain  tone  is  common  to  three  chords  there  will  be 
six  combinations.  The  2d,  4th,  and  jth  tones  admit  but  two  com- 
binations. 

The  same  exercise  should  be  written  in  B-flat  and  A-flat,  adding 
the  root-notes  in  the  base  afterwards. 

These  combinations  should  also  be  re-arranged  in  two  other 
positions,  the  fundamental  part  remaining  the  same. 

One  example  is  given  as  illustration : 


All  these  combinations  might 
be  performed  simultaneously  on 

various    instruments   with   good 

Ex.  67.  f9      .  .,  effect.     Observe    that    the   same 

base    accompanies    each    of   the 
arrangements,  i,  2,  or  3. 


Chapter  IX. 


HARMONIZATION   OF  A   GIVEN  THEME. 

APPLICATION  OF  THEORETICAL  PRINCIPLES. 


difference  between  Harmonization  and  Harmonic  Progres- 
sion  should  now  be  understood. 

Harmonic  Progression  relates  to  the  manner  of  moving  from 
one  chord  to  another,  without  regard  to  melody,  as  in  previous 
examples.  The  order  in  which  the  chords  iucceed  one  another 
depends  upon  the  natural  order  in  which  they  occur  in  the  chart. 

Harmonization  presupposes  a  melody,  which  is  accompanied 
with  certain  chords,  following  the  theme  in  proper  succession.  In 
Harmonization  one  chord  may  be  followed  by  any  other  chord, 
provided  they  succeed  each  other  correctly.  No  chord  can  right- 


GOODRICH'S  ANALYTICAL  HARMONY.  39 

fully  be  forbidden  to  follow  another.  The  only  present  restriction 
is,  that  the  student  is  confined  to  the  use  of  chords  between  which 
there  is  at  least  one  connecting  note.  In  harmonizing  a  given 
melody  the  progressions  must  be  written  according  to  previous 
directions,  the  object  being  to  make  the  theoretical  information  prac- 
tical and  available.  For  instance,  if  g  and  a,  in  the  Key  of  C  are 
melody  notes,  any  chord  that  contains  g  may  accompany  the  first 
note  and  any  chord  containing  a  may  accompany  the  second  note. 
In  other  words,  the  melodic  note  may  be  root,  third  or  fifth  of  the 
harmony.  If  the  G  chord  is  selected  to  harmonize  g  and  the  D  chord 
to  harmonize  a,  the  question  arises,  do  these  chords  succeed  each 

other  properly  ?     See  example :      Ex-  68-  P^— d — &—  Between 


the  two  chords  there  is  a  note  in  common,  d,  but  this  connecting 
note  does  not  remain  in  the  same  voice-part.  In  skipping  from  d  to 
f,  it  compels  the  contralto  to  skip  from  b  to  d.  This  error  is  caused 
by  the  ascending  melody ;  whereas,  if  the  progression  from  the  G 
chord  to  the  D  chord  is  correctly  written,  the  voices  will  descend : 


Ex-  &9'  r/m~d~  (Compare  this  with  the  previous  example.) 


The  student  should  understand  that  he  is  not  forbidden  to  go  from 
the  G  to  the  D  chord,  unless  the  movement  of  the  melody  is  contrary 
to  the  correct  progression  of  the  chord. 

Another  harmonization  is  therefore  made.  Select  the  C  and  F 
chords  as  accompaniment  : 


Ex-  7°-  SJF==     This  is  correct. 


The  minor  chords  of  E  and  A  would  be  equally  correct : 
Ex.  71. 


A  simple  diatonic  theme  will  now  be  presented  for  harmonization 
in  accordance  with  present  information.  This  theme  is  continuous, 
and  there  must  be  at  least  one  connecting  note  throughout.  The 
note  of  connection  may  occur  in  any  of  the  upper  parts,  soprano, 


40  GOODRICH'S  ANALYTICAL  HARMONY. 

mezzo-soprano,  or  contralto,  but  every  chord  must  be  connected  in 
some  manner  with  its  antecedent  and  consequent : 


Ex.  72. 


Do   si     la    si      do    —    re    —      mi  fa    mi    re      —    do    —    si       do. 

i       i : 1 — i — ^          ^        i  *,    /g    ~^.      ;^c 

— " — ^ — 75 — 


To  facilitate  the  work  of  harmonization  it  will  be  necessary  to 
prepare  a  chart  of  the  concords  in  B-flat,  arranged  under  each  of  the 
seven  notes  of  the  scale  : 


CHART    FOR    B-FLAT. 

^  —  . 
^~~»  sol 


The  first  note  in  the  theme  is  tonic. 

According  to  the  chart  there  is  a  choice  of  three  chords  in  har- 
monizing this  note,  but  owing  to  the  direction  of  the  theme,  only 
one  will  serve  the  present  purpose. 

The  next  note  is  «'.  (See  last  measure  of  chart.)  Either  of  the 
chords  (D  or  F)  may  follow  the  first  chord ;  but  as  there  must  be  a 
connection  between  the  second  and  third,  as  well  as  between  the  first 
and  second  chords,  only  one  of  the  chords  at  j/  can  be  used. 

The  harmonization  of  la  will  determine  the  correctness  of  the 
second  chord. 

Care  must  be  taken  to  write  the  connecting  notes  in  the  same 
voice-parts. 

If  the  first  three  chords  are  correctly  written  there  will  be  no 
difficulty  in  harmonizing  the  two  following,  for  whatever  is  correct 
in  descending  will  be  correct  in  ascending,  the  order  being  merely 
reversed. 

Whenever  a  note  is  tied  in  the  melody,  the  harmony  must  change 
to  some  other  chord  containing  that  note.  In  such  instances  the 
melody  is  the  connection.  The  tied  notes  in  the  soprano  part  make 
it  possible  to  ascend  without  transgressing  present  rules. 

After  completing  the  upper  parts,  the  base  is  to  be  added  by  giv- 
ing to  that  part  the  Foot  of  each  chord,  whatever  may  be  its  position. 
*  *  * 

(The  harmonized  theme  may  be  found  in  the  key,  but  this  must 
not  be  consulted  except  for  comparison,  after  the  student's  example 
has  been  completed.) 

«-The  syllables  in  the  theme  correspond  to  those  in  the  chart. 


GOODRICH'S  ANALYTICAL  HARMONY.  41 

Every  example  should  be  performed  upon  a  piano  or  organ  in 
moderate  tempo,  and  with  sufficient  distinctness  to  produce  the 
full  harmonic  effect,  care  being  exercised  to  sound  the  tones  of  each 
chord  simultaneously.  Such  practice  improves  the  student  in  sight- 
reading,  and  what  is  more  important,  enables  him  to  judge  the  effect 
of  chord  progressions,  and  eventually  to  comprehend  harmonic  suc- 
cessions by  sight  alone. 

The  importance  of  these  auricular  exercises  has  already  been  set 
forth  in  the  author's  Complete  Musical  Analysis  as  a  necessary  feature 
of  all  thorough  musical  education. 

Notes  in  the  theme  marked  +  admit  a  choice  of  harmony.  For 
instance,  the  6th  chord  may  be  either  G-minor,  or  E-flat.  It  is  not 
well  to  burden  the  student's  mird  with  cadence  forms  when  the  only 
object  at  present  is  to  acquire  the  ability  to  handle  chords  correctly. 

Therefore  if  a  pupil  ends  the  harmonization  like  this : 


Ex.  74. 


it  may  be  considered  as  correct  as  an  example  ending  with  a  regular 
cadence,  thus : 


Ex.  75. 


The  theme  for  harmonization  is  to  be  transposed  into  C,  D,  and  A. 
In  each  instance  a  chart  should  be  prepared  as  a  means  of  supplying 
the  accompanying  harmonies.  No  two  chords  are  to  appear  in  suc- 
cession in  the  same  position,  and  the  base  must  not  skip  more  than 
a  5th. 

For  a  separate  lesson  a  more  extended  theme  is  presented,  to  be 
harmonized  according  to  the  same  principles  and  directions.  This 
also  is  to  be  transposed. 

It  would  be  well  in  writing  these  examples  not  to  exceed  the 


compass  of  four  octaves :       EX.  76. 


GOODRICH  S    ANALYTICAL    HARMONY. 


Ex.  77. 


^a 


THEME    FOR    HARMONIZATION. 

Mi      re  do  si       la       si 


4= 


SE 


mi     fa^. ^        sol        fa^  mi      re  _^         do  si  la        si  do 

t^t  fi^ ^?          \  fS ^^ ^Q 

-rz?      t—  p-~  r-^^ 


Chapter  X. 


HARMONIZATION  OF  MELODIC  SKIPS  OF  A 

THIRD. 

HERETOFORE  each  voice-part  has  moved  alphabetically.     Me- 
lodic skips  of  a  third  will  now  be  introduced. 
The  directions  are  simple. 

A  skip  of  a  3d,  major  or  minor,  may  be  harmonized  with  any 
chord  containing  both  notes  of  the  skip,  either  ascending  or  descend- 
ing. 


Suppose  these  tones  occur  in  a  melody :  Ex-  78-  L 


Any  chord  containing  a  and  c  will  accompany  this  interval,  both 
tones  being  a  part  of  the  same  chord.  Mention  chords  containing 
a  and  c. 

The  result  is  merely  a  re-arrangement  of  any  chord  containing 
a  and  c. 

Two  harmonizations  should  be  attempted.     *     *    * 

The  simplest  plan  is  to  consider  both  tones  of  the  skip  as  parts 
of  one  chord,  the  fundamental  remaining  unchanged. 

Any  chord  written  in  its  different  positions  will  illustrate  this : 


Ex.  79. 


None  of  the  previous  directions  are  violated,  as  the  harmony  remains 
the  same  throughout. 


GOODRICH'S  ANALYTICAL  HARMONY. 


43 


The  melodic  notes  g,  b,  d,  indicate  the  chord  of  G,  for  that  chord 
is  composed  of  those  notes.  As  only  one  chord  is  involved  in 
the  harmonization  of  a  skip,  the  rule  for  connecting  notes  does  not 
apply. 

The  development  of  Ex.  78  is  presented  for  comparison,  prepar- 
atory to  the  harmonization  of  a  theme  containing  skips : 


Ex.  80. 


Root. 


Root. 


Both  examples  are  correct.     The  notes  of  the  skip  are  the  3d  and 
5th  of  the  F  chord  and  root  and  3d  of  the  A-minor  chord. 

The  pupil  should  attempt  to  harmonize  the  following  theme  in 
skips,  according  to  the  directions  contained  in  this  chapter  : 


THEME  IN  SKIPS. 


Ex.  81. 

-  y  4  — 

~3?  — 

s  

H  «•  — 

Ug,  __  -j^  1 

-(&2  —  ^-^* 

—  «•  

^        1 

The  first  skip  must  be  accompanied  with  a  chord  containing  both 
tones  of  the  skip  (E  and  g).  At  the  second  g  the  chord  is  to  be 
changed  to  one  containing  g  and  b.  Therefore  the  change  of  har- 
mony must  have  a  connecting  note,  and  be  written  according  to 
previous  rules  of  chord  progression.  The  two  g's  above  serve  as 
connecting  notes.  The  entire  second  measure  is  to  be  accompanied 
with  the  same  chord,  because  the  harmony  can  not  be  changed 
during  the  progress  of  a  skip. 

G  and  a  in  the  third  measure  are  to  be  accompanied  with  two 
chords  having  a  connecting  tone  between  them  and  in  the  same 
voice-part,  this  being  a  regular  harmonic  progression.  A,  descend- 
ing to/",  is  to  be  accompanied  by  one  chord,  and  f  descending  to  d, 
by  another  chord.  D  and  b  require  still  a  different  chord  (containing 
those  tones),  and  from  b  to  c  at  the  close  the  harmony  changes  and 
there  must  be  a  connecting  note. 

The  chart  for  C  may  be  consulted  in  determining  how  many 
chords  will  accompany  each  tone  of  the  scale,  and  what  chords  em- 
brace the  tones  of  a  skip,  either  ascending  or  descending.  Remem- 


44 


GOODRICH'S  ANALYTICAL  HARMONY. 


ber  that  both  tones  of  a  skip  must  be  harmonized  with  the  same 
chord.  The  accompanying  chord  in  such  cases  merely  changes  its 
position,  not  its  name  or  root.  The  harmonization  of  the  last  theme 
will  be  found  in  the  key.  Transpose  the  skipping  theme  into  B-flal 
and  D-flat  and  harmonize  accordingly.  A  theme  embracing  every 
concord  in  the  scale  is  presented  for  harmonization. 


Ex.  82. 

-j  — 

-1 

*           (?5 

r^~ 

-ptf-2   - 

^  —  s 

?  ^ 

i       !•     ' 

* 

»/ 

p 

9  

^~^ 

^ 

rr  f  ?L-I 

- 

—  !  — 

1  

-P  —  '  ' 

1         i 

Whenever  the  melody  moves  alphabetically  the  harmony 
changes,  and  there  must  be  a  note  of  connection.  Where  the 
melody  skips,  the  same  harmony  is  to  be  continued. 

Transpose  this  theme  into  A  and  F. 


Chapter  XL 


I 


HARMONIZATION  OF  MELODIC  SKIPS 
OF  A  FOURTH. 

N  changing  the  position  of  a  chord,  a  skip  of  a  4th  will  appear  in 


some  voice-part : 


Ex.  83. 


The  soprano  and  mezzo- 


r 


soprano  skip  a  3d  each,  but  the  contralto  leaps  a  4th.    The  notes 
of  a  chord  written  melodically  will  illustrate  this : 


Ex.  84*.  bfct 


GOODRICH 'S    ANALYTICAL    HARMONY. 


45 


These  notes  comprise  and  consequently  indicate  the  F  chord,  from 
w'nch  we  may  conclude  that  F  is  the  accompanying  harmony : 


Ex.  84^. 


The  skip  of  a  4th  occurs  first  in  the  mezzo-soprano,  then  in  the 
contralto,  and  finally  in  the  soprano  part. 

Directions. 

The  chord  that  naturally  harmonizes  the  skip  of  a  4th  either 
ascending  or  descending,  contains  both  tones  of  the  skip.  The 
student  should  examine  the  six  concords  in  G  in  order  to  discover 


what  chord  contains  this  interval :     EX.  85. 


The  con- 


cord containing  b  and  e  forms  the  accompaniment,  the  base  remain- 
ing unchanged.  The  accompanying  harmony  merely  assumes  differ- 
ent positions.  The  following  short,  continuous  theme,  proceeding 
mostly  by  leaps  of  a  4th,  is  to  be  harmonized  as  already  explained : 


Ex.  S6. 


7wb        1    /i 

(5? 

^ 

^^^ 

C^       I 

^^ 

-ffir  J 

<v 

i 

1  — 

—  H1- 

d 

—  ^ 

r^— 

There  is  scarcely  a  possibility  of  failure  in  harmonizing  this  cor- 
rectly, provided  the  previous  explanations  have  been  read  attentively. 
The  first  two  notes  of  the  theme  constitute  two  thirds  of  the  accom- 
panying chord.  The  next  skip,  a  to  d,  is  similarly  indicated.  When 
there  is  no  skip,  the  harmony  changes.  In  such  instances  the  con- 
necting-note principle  must  be  applied. 

Transpose  the  theme  into  G  and  A-flat  and  harmonize  in  four 
parts.  *  *  * 

The  following  melody  consists  of  a  sequence  of  thirds  and  fourths 
and  should  be  so  harmonized  as  to  include  every  concord  in  the  scale 


46 


GOODRICH'S  ANALYTICAL  HARMONV. 


of  A.     When  the  melody  moves  alphabetically,  the  rules  of  pi  ogres- 
sion  are  to  be  observed : 


Ex.  87. 


V      .ft 

fi/       1           i 

1                  2 

f?         '      /n 

^     ,    "    ^  1 

Transpose  into  B-flat  and  C 

The  next  theme  is  similar  to  the  others,  and  presents  no  new 
difficulties : 


"a~ 


Ex.  88. 


=2± 


i        1        ' 

1                            ^-1        /^ 

1 

1      ' 

1              1             p*^ 

-J          ^r          j                 ^^ 

'. 

No  other  melodies  are  to  be  harmonized  at  present,  as  each  theme 
has  been  expressly  contrived  for  the  situation  in  which  it  occurs,  m 
order  to  illustrate  each  new  subject  without  violating  the  directions- 
prescribed  for  the  student's  guidance. 


ANALYTICAL,   HARMONY. 


PART  III 


Chapter  XII. 


FORBIDDEN  PROGRESSIONS. 


is  a  somewhat  ungracious  subject  to  discuss,  for  many  of 
JL  the  forbidden  progressions  are  freely  used  by  modern  com- 
posers. One  of  the  most  objectionable  of  these  is  the  parallel 
movement  by  fifths.  This  prohibition  does  not  apply  to  the  appear- 
ance of  a  5th  in  each  chord,  but  to  the  parallel  movement  of  any 
two  voice-parts^at  the  distance  of  a  5th,  thus : 


Ex.  89. 


The  contralto  and  soprano  parts  move  in  parallel  directions  at  the 
distance  of  a  5th.  All  such  progressions  are  forbidden,  and  rightly 
so,  for  the  effect  is  certainly  abrupt  and  unsatisfactory.  Parallel 
5ths  between  the  base  and  mezzo-soprano  parts  are  here  shown : 


Ex.  go. 


* 


m 


These  are  not  so  prominent  on  account  of  the  holding-tone  above, 
but  they  are  false  and  must  be  condemned. 

*The  interval  here  called  a  sth  is  really  a  I2th,  but  they  are  theoretically  synonymous 
The  same  may  be  said  of  octaves,  which  are  so  called  even  though  they  are  two  or  three 
octaves  apart. 


GOODRICH'S   ANALYTICAL    HARMONY. 


Consecutive  5ths  are  liable  to  occur  between  any  two  parts,  and 
these  inaccuracies  must  be  detected.  In  the  next  example  the  base 
and  contralto  parts  move  at  the  distance  of  a  5th,  and  the  result  is 
a  grammatical  error : 


Ex.  91. 


Though  there  are  situations  in  which  this  might  be  used  with  im- 
punity, it  is  better  for  the  student  to  avoid  all  these  transgressions 
until  full  independence  from  teacher  and  text-book  has  been  achieved. 
As  almost  every  chord  contains  a  5th,  it  is  only  necessary  to 
observe  that  the  two  voice-parts  which  produce  a  5th  do  not  move 
together  at  the  same  distance.  In  the  following  example  there  is  i 
5th  in  every  chord,  yet  no  false  progression  appears : 


Ex.  92. 


Each  chord  should  be  examined  in  order  to  locate  the  5ths,  and  to 
observe  the  fact  that  no  two  parts  move  by  parallel  5ths. 

By  observing  the  previous  directions  no  false  progressions  will 
result.  For  this  reason  the  author  has  said  little  about  these  re- 
strictions. 

Another  parallel  movement  forbidden  by  theorists  is  that  of  the 
octave,  or  isth. 

Such  an  example  is  presented: 


Ex.  93. 


E£ 


GOODRICH'S  ANALYTICAL  HARMONY. 


49 


The  parallel  lines  show  the  consecutive  octaves  between  the  base 
and  one  of  the  upper  parts.  The  progression  from  the  F-major  to 
the  G-minor  chords,  in  the  first  measure,  is  especially  objectionable, 
as  it  involves  parallel  5ths  in  addition  to  the  octaves.  Such  defects 
usually  result  from  a  similar  movement  of  the  parts,  causing  a  5th 
or  an  octave  to  move  to  another  5th  or  octave.  These  errors  may 
be  avoided  by  employing  contrary  or  oblique  motion.  Following 
are  examples  of  these  three  kinds  of  movement : 


Ex.  94. 


Parallel. 


^~v  — 


Contrary. 
g     fS 


Oblique. 


feE 


1 


The  first  example  is  bad ;  the  other  two  are  good.     Progressions  like 
these  may  be  included  among  oblique  movements : 


Ex.  95. 


Though  the  parts  apparently  move  in  the  same  direction,  the  con- 
necting note  prevents  false  progressions.  The  base  in  such  instances 
may  descend  as  at  (a)  and  (b),  or  ascend  as  at  (c). 

With  a  view  to  avoiding  ungrammatical  progressions  the  student 
should  observe  particularly  that  the  parts  producing  a  5th  or  octave 
do  not  move  the  same  distance  in  a  parallel  direction : 


Ex.  96. 


The  base  and  soprano  are  an  octave  (or  isth)  apart  in  the  C  chord; 
but  in  the  G  chord  they  are  a  loth  (or  lyth)  apart.  In  other  words, 
the  base  part  descends  a  4th,  the  soprano  a  2d.  The  connecting  note 


5O  GOODRICH'S  ANALYTICAL  HARMONY. 

in  the  mezzo-soprano  gives  to  the  upper  parts  the  effect  of  oblique 
motion,  and  in  such  instances  the  base  may  ascend  a  5th  or  descend  a 
4th. 

A  distinction  is  to  be  made  between  consecutive  octaves  and  uni- 
sons. The  following  are  not  intended  to  come  within  the  scope  of 
prohibited  passages : 


Ex.  97. 


At  (a)  the  base  is  doubled  below;  at  (b)  the  melody  is  doubled  above. 

These  are  mere  re-enforcements  or  duplications  of  an  extreme 
part,  and  as  there  is  no  harmony  between  these  unisons  they  are 
perfectly  correct. 

As  the  other  proscribed  relations  and  progressions  are  not  liable 
to  occur  at  present,  their  discussion  is  left  to  a  future  chapter. 

The  errors  in  the  following  exercise  should  be  corrected,  and  the 
theme  may  be  altered  in  order  to  keep  within  the  limits  of  present 
information.  Preserve  the  same  base  : 


Ex.  98 


i 


P=3=j=3: 


EBB 

*&   .   '*•'  W*-.—- -  J 


TT^C 


GOODRICH 'S    ANALYTICAL    HARMONY. 


Chapter  XIII. 


THIRTY  HARMONIC  PROGRESSIONS  IN  A 

MAJOR  KEY,  WITH  AND  WITHOUT 

CONNECTING  NOTES. 


previous  harmonic  progressions  have  been  confined  to  chords 
having  at  least  one  note  in  common.  But  as  these  exclude  all 
such  chord  movements  as  from  C  to  D  and  D  to  E,  our  field  of  oper- 
ations must  be  enlarged  so  as  to  include  all  combinations  in  the  key. 
The  object  is  to  make  every  possible  progression  ;  and  to  do  this,  it 
is  necessary  to  have  a  systematic  method  for  determining  the  order 
of  these  combinations.  We  begin  with  the  C  chord  and  progress 
from  this  to  each  of  the  other  five  concords  in  regular  order.  Next, 
begin  with  the  D  chord  and  move  to  every  other  chord  according  to 
the  Harmonic  Index  : 


Ex.  99. 


1         2 


HARMONIC  INDEX. 
45  678         9        10         11 


'<o   14    . 1|S 


This  diagram  shows  the  order  of  the  progressions  fundamentally. 
Each  base  note  is  a  root,  and  represents  the  concord  founded  upon 
that  note.  The  theory  will  now  be  explained. 

There  is  no  note  in  the  C  chord   that  occurs  in  the  D  chord. 
They  can  not  succeed  each  other  in  this  manner : 


Ex.  too. 


for  it  involves  both  parallel  fifths  and  octaves. 


1 


GOODRICH'S  ANALYTICAL,  HARMO>  -. 


As  the  base  is  to  move  from  root  to  root,  the  upper  parts  must  be 
altered.  Contrary  movement  will  obviate  the  difficulty.  Begin  again, 
with  the  C  chord,  with  the  base  as  a  foundation : 


Ex.  101. 


J 

^£j  * 


The  first  note  of  the  D  chord  below  c  (in  the  soprano)  is  a.     This 
will  indicate  that  the  D  chord  is  to  appear  with  a  uppermost.     The 
note  of  the  D  chord  next  below  g  is  /,  and  next  below  e  is  d. 
The  progression  will  thus  appear  in  correct  form : 


Ex.  102. 


The  base  moves  from  root  to  root,  while  the  e  and  g  descend  to  d 
and  /.     This  is  not  unusual : 


Ex.  103. 


The  interval  of  a  fifth  in  the  first  chord  is  followed  by  a  third  in  the 
second  chord  and  a  false  progression  is  thus  avoided.  But  the  skip 
from  c  to  a  in  the  upper  part  must  be  explained  as  the  result  of 
necessity.  It  prevents  consecutive  octaves  and  supplies  the  remain- 
ing note  of  the  D  chord.  (See  examples  100  and  102.) 

Another  ameliorating  circumstance  is  this :  all  the  other  voice- 
parts  move  alphabetically.  Even  the  base,  which  has  hitherto  skipped 
a  third,  fourth  or  fifth,  here  moves  but  a  second.  The  duplicated 
root-note  is  therefore  the  only  one  that  skips. 

From  the  foregoing  may  be  deduced  the  principle  that  when 
there  is  no  connecting  note  between  two  chords,  the  treble  parts 
must  move  in  an  opposite  direction  to  that  of  the  base.  To  be  still 
more  explicit,  when  the  base  ascends  a  second  the  other  parts  descend  ; 
and  when  tne  base  descends  a  second  the  other  parts  must  ascend- 


GOODRICH'S  ANALYTICAL  HARMONY. 


53 


When  the  base  ascends  or  descends  fundamentally  a  major  or  a 
minor  second  there  will  be  no  connecting  note  between  the  two 
chords.  This  is  always  true  of  concords. 

Xole.  Some  writers  consider  the  progression  in  Ex.  102  incorrect  on 
account  of  "  hidden  fifths  "  between  the  soprano  and  contralto  parts.  These 
may  he  avoided  by  resolving  the  e  up  to  f,  instead  of  down  to  d.  But  com- 
posers seldom  concern  themselves  with  these  prohibitions,  as  the  following 
extract  shows : 

(From  the  Landing  of  the  Pilgrims.    By  G.  W.  Chadwick.) 

Unaccompanied. 

t- 


Ex.  104. 


Aye,  call     it      ho  -  ly  ground. 


The  author  has,  therefore,  no  hesitancy  in  recommending  the  progression  as 
given  in  Ex.  102. 

Each  progression  is  to  be  noted  in  three  positions,  the  base  being 

the  same : 

1. 

3 


Ex.  105. 


it 


Complete  the  example.     This  is  the  first  progression.     *    *    * 

No.  2  is  from  C  to  E.  This  is  not  new,  for  there  are  two  con- 
necting links.  These  are  to  be  retained  in  the  same  voice-parts  as 
formerly.  The  intention  is  to  supplement,  not  to  contradict  any 
of  our  previously  acquired  principles. 

Write  three  arrangements  of  this  progression  and  number  it  2. 
N"o.  3  is  from  C  to  f. 

Each  progression  is  to  be  arranged  in  three  positions  and  num- 
bered according  to  the  index.  In  the  fifth  progression  the  base 
should  descend  a  third  rather  than  ascend  a  sixth.  The  upper  parts 
present  no  new  difficulties : 


Ex.  1 06. 


54 


GOODRICH 'S    ANALYTICAL    HARMONY. 


These  are  all  the  progressions  that  can  be  made  from  the  initial 
chord ;  therefore  begin  with  D,  No.  6.  Here  the  base  moves  up  a 
ad,  and  there  is  no  note  in  common.  Use  the  D  chord  in  its  first 
position  and  write  the  base,  D  to  E.  The  soprano  moves  to  that 
note  of  the  E  chord  next  below  the  5th  of  the  D  chord.  This  fixes 
the  position  of  the  second  chord : 


Ex.  107. 


Therefore  f  goes  to  <?,  while  d  skips  to  b,  the  5th  of  the  E  chord. 
Write  this  in  three  positions.  In  progressing  from  D  to  F,  D  to 
G,  and  D  to  A,  no  obstacles  will  appear.  The  imperfect  triad  on  B 
being  omitted,  the  next  progression  is  from  D  to  C. 

Do  not  move  the  base-part  up  a  seventh,  but  down  a  second,  so 
as  not  to  violate  the  rule,  that  the  base  must  not  skip  more  than  a 
fifth.  As  the  base  descends  a  second  the  other  parts  must  ascend. 
The  soprano  note  a  goes  up  to  that  interval  of  the  C  chord  nearest 
above  a.  In  other  words,  we  reverse  the  first  progression,  C  to  D. 
After  writing  so  much,  it  is  best  to  indicate  the  soprano  part  first, 
because  this  will  decide  the  position  of  the  second  chord : 


Ex.  108. 


After  this  the  other  notes  are  easily  supplied.  Include  the  re- 
arrangements. For  the  eleventh  progression  see  index.  Write  the 
base,  then  the  first  position  of  the  E  chord.  The  .soprano  part  of  the 
second  chord  is  to  be  indicated  next  and  the  middle  parts  after- 
wards, as  heretofore  explained. 

The  fundamental  order  in  which  the  chords  are  combined  into 
thirty  progressions  is  so  plainly  shown  by  the  index  that  the  student 
can  accomplish  this  task  without  further  aid  than  the  recapitulation 
of  our  few  governing  principles. 

Chord  progressions  may  be  divided  into  two  classes : 

ist,  with  connecting  notes  ; 

2d,  without  connecting  notes. 


GOODRICH'S  ANALYTICAL  HARMONY.  55 

The  first  includes  those  progressions  in  which  the  base  moves 
t_p  or  down  a  30!,  4th,  or  5th. 

Whenever  the  base  moves  up  or  down  a  2d  (whole  or  half  step) 
chere  will  be  no  connecting  note.  Contrary  movement  is  then  im- 
perative. 

When  there  is  a  note  in  common,  the  base  may  move  in  similar 
or  contrary  motion.  The  only  exception  to  this  is,  that  a  leap  of  a 
6th  in  the  base  part  should  not  be  substituted  for  that  of  a  3d.  The 
/atter  is  preferable.  Only  one  note  of  a  chord  can  skip  when  the 
harmony  changes ;  the  others  must  move  by  regular  degrees.  In 
harmonizing  skips,  all  the  parts  leap  except  the  base,  which  remains 
stationary,  as  the  harmony  does  not  change.  It  would  also  be  well 
to  remember  for  the  present  that  no  two  chords  follow  each  other  in 
the  same  position,  and  the  base  is  to  be  given  the  root  of  each  chord. 

These  directions  are  plain,  and  if  followed,  all  difficulties  will  be 
overcome.  At  least  such  is  the  author's  experience  in  teaching  this 
system  during  the  past  twenty-one  years. 

The  thirty  progressions  here  outlined  are  all  that  can  be  made 
from  the  six  concords  in  a  normal  scale.  Then  by  re-arranging  each 
progression  in  two  other  positions  we  literally  exhaust  the  subject. 

Those  who  expect  to  become  good  harmonists  should  write  out 
these  thirty  progressions  in  at  least  six  other  major  scales,  namely  : 
£),  E,  F-sharp,  B-flat,  A-flat,  and  G-flat.  Each  exercise  is  to  be  per- 
formed after  it  is  written.  The  solution  of  this  lesson  will  be  found 
in  the  Key. 


Chanter  XIV. 


HARMONIZATION  OF  THEMES  WITH  AND  WITH- 
OUT CONNECTING  NOTES. 


CHORD  RELATIONS. 


THE  student  is  now  prepared  to  harmonize  any  melody,  provided 
it  contains  no  appoggiaturas,  passing-notes,  or  chromatic  alter- 
ations.    So  long  as  the  melody  note  is  considered  as  part  of  a  con- 
cord, it  will  present  no  difficulties. 


56  GOODRICH;S   ANALYTICAL   HARMONY. 

Melodic  skips  of  a  3d  and  4th  are  also  understood.  To  be  precise, 
the  student  is  familiar  with  the  thirty  harmonic  progressions  in  every 
key,  each  in  three  positions.  This,  together  with  the  skips,  will  an- 
swer all  present  purposes. 

Begin  with  the  descending  scale : 


Ex.  109. 


As  our  material  consists  of  but  six  chords,  the  chart  should  be  dis- 
pensed with.  Endeavor  to  perceive,  mentally,  the  chords  that  ac- 
company each  note  and  the  three  close  positions  of  each  chord. 
Then  choose  that  one  which  will  form  a  correct  progression.  The 
example  is  to  be  harmonized  with  unrelated,  as  well  as  with  related 
chords ;  i.  e.,  with  and  without  connecting  notes.  Begin  with  any 
chord  containing  g;  the  G-major  or  E-minor  chord  will,  however,  be 
preferable.  When  there  is  a  connecting  note,  keep  it  in  the  same 
voice-part ;  when  the  chords  are  unrelated,  move  the  upper  parts  in 
a  contrary  direction  to  that  of  the  base.  Begin  and  end  with  any 
chord  that  forms  a  correct  progression  with  its  consequent  and  ante- 
cedent. *  *  * 

As  it  is  impossible  to  write  a  progression  here  not  already  written 
in  the  previous  chapter,  the  author  concludes  that  the  harmonization 
of  Ex.  109  will  be  successfully  accomplished.  When  it  is  completed 
the  same  melodic  notes  should  be  copied,  and  another  entirely  differ- 
ent arrangement  made.  Every  corresponding-note  of  the  scale  pas- 
sage is  to  be  accompanied  with  a  different  chord.  A  brief  prepara- 
tory exercise  will  serve  as  illustration  : 


Ex.  no. 


-I — *- 


Both  examples  are  equally  correct.  That  some  of  these  chord  pn> 
gressions  sound  inharmonious  is  merely  because  we  have  become 
pocustomed  to  certain  cadence-harmonies.  It  is  certainly  no  fault  of 
the  progressions,  for  they  have  been  used  by  all  classic  composers. 
Our  present  object,  however,  is  to  acquire  the  art  of  managing  chords 
correctly  and  systematically.  After  this  is  accomplished  the  student 


GOODRICH 'S    ANALYTICAL    HARMONY. 


57 


may  harmonize  a  theme  with  those  chords  that  sound  the  most  agree- 
able, or  that  represent  a  melodic  idea  to  the  best  advantage.  The 
two  harmonizations  of  Ex.  109  will  be  found  in  the  Key. 

A  section  of  melody  is  presented  for  harmonization  in  two  differ- 
ent ways  : 


Ex.  in. 


It  might  be  better  to  harmonize  one  note  at  a  time  in  each  example, 
in  order  to  produce  these  different  arrangements  readily.  When 
completed,  each  copy  should  be  examined  critically  for  the  purpose 
of  detecting  possible  errors.  Transpose  the  last  example  into  A,  C, 
D,  and  E-flat. 

This  part  of  the  chapter  closes  with  a  more  extended  theme, 
which  is  designed  to  include  all  the  principles  of  progression  and 
harmonization  thus  far  explained.  It  should  begin  and  end  in  D  : 


Ex.  112. 


>— .&- 


~: 


1 


-&• <9- 


fz« 

1              J<? 

_.                                         ,'-*' 

f3               & 

^ 

^ 

I 

i 

!  

J  —  1  —  r- 

4— 

^ 

Transpose  to  B  and  E. 

CHORD  RELATIONS. 

Attention  is  here  directed  to  the  natural  connection  and  relation 
of  consonant  chords  in  a  given  scale. 

In  proceeding  by  fifths  there  will  always  be  one  connecting  note 
in  the  upper  part,  which  will  appear  alternately  as  5th  of  one  chord 
and  root  of  another  : 


Ex.  113. 


-&- 


The  progression  of  the  base  from  d  down  a  fourth  to  a  is  the  same  in 
harmonic  effect  as  the  upper  fifth.  This  latter  would  cause  the  base 
to  ascend  too  high,  and  the  lower  fourth  is  therefore  substituted- 
These  progressions  by  fifths  have  a  receding  tendency,  as  from  sub- 
dominant  to  tonic. 


GOODRICH'S  ANALYTICAL  HARMONY. 


The  progression  by  fourths  is  the  same  in  regard  to  connection 
of  tones : 


Ex.  114. 


In  both  instances  there  is  one  connecting  link  throughout.  But  the 
effect  of  this  latter  is  almost  opposite  to  that  of  the  former,  for  in 
place  of  a  receding  tendency  this  has  a  progressive,  advancing  ten- 
dency. Compare  (a)  with  (b)  in  the  following  example : 


Ex.  115. 


One  is  mild  and  undecided,  the  other  is  strong  and  of  a  transitional 
nature. 

The  progressions  by  thirds  present  the  greatest  number  of  tone 
connections.  This  is  true  in  ascending  and  in  descending  movements. 
The  student  should  supply  the  upper  harmony  in  the  two  succeeding 
examples : 


Ex.  116. 


Roots. 

The  descending  movement  will  result  in  this  way :  dominant  and 
relative  minor ;  tonic  and  relative  minor ;  sub-dominant  and  relative 
minor.  Also  add  the  upper  harmony  to  this  ascending  progression 
by  thirds : 


Ex.  117. 


Roots. 


In  both  examples  the  base  notes  represent  roots,  therefore  the  chords 
are  added  according  to  the  principles  of  harmonic  progression.  In 
each  example  there  will  be  two  connecting  notes,  so  long  as  the 
relation  of  parallel  thirds  is  maintained.  In  the  last  example  the 
former  order  as  to  major  and  relative  minor  is  reversed,  and  each 


GOODRICH'S   ANALYTICAL    HARMONY. 


minor  chord  is  followed  by  its  relative  major.  The  ascending  move- 
ment is  more  natural  and  progressive.  The  parallel  progressions  are 
intimately  related  and  connected,  and  in  a  future  chapter  the  prin- 
ciples of  chord  relation  will  be  applied  to  keys  and  modes  as  well  as 
to  chords  and  progressions. 

With  regard  to  progressions  by  seconds  they  have  no  actual  con- 
nection, and  no  apparent  relationship  with  one  another,  excepting  by 
means  of  an  intervening  chord.  This  may  be  observed  here  by  con- 
sidering each  root  as  a  tonic,  and  associating  each  chord  with  the 
signature  of  that  key  : 


Ex.  118. 


The  first  (B-minor)  has  two  sharps ;  the  second  has  four  sharps ;  the 
third  two,  and  the  fourth  has  four.  But  where  smoothness  and  con- 
nection are  not  desirable,  these  progressions  serve  a  purpose,  for  they 
are  bold  and  disconnected.  The  next  exercise  begins  and  ends  in 
A-major  and  the  student  may  add  the  proper  harmonies,  considering 
each  base  as  a  root : 


Ex.  119. 


1 


From  the  foregoing  we  may  deduce  the  following  simple  formula : 
When  the  base  moves  a  fourth  or  a  fifth  up  or  down,  there  will  be 
one  connecting  note  above  ;  when  the  base  moves  a  major  or  a  minor 
third  up  or  down,  there  will  be  two  connecting  notes ;  when  the 
base  ascends  or  descends  a  whole  or  a  half  step,  there  will  be  no  con- 
necting link  between  the  chords,  and  the  upper  harmony  must  in 
such  instances  move  in  an  opposite  direction.  In  all  the  elementary 
exercises  the  root  of  each  chord  is  to  appear  in  the  base  as  funda- 
mental. 

As  a  sixth  is  an  inversion  of  a  third  it  is  not  included  among 
these  fundamental  progressions,  because  the  skip  of  a  third  in  the 
base  is  nearly  always  preferable  to  the  skip  of  a  sixth. 

Transpose  and  re-arrange  the  last  seven  examples  until  the  prin- 
ciple is  well  understood.  Also  listen  to  the  different  effects  of  these 
various  progressions. 


GOODRICH  S  ANALYTICAL   HARMONY. 


Chapter  XV. 


ANOTHER  METHOD  OF  HARMONIZING  MELODIC 
SKIPS  OF  A  THIRD. 


IN   progressing  from  one  chord  to  another  between  which  there 
was  no  connecting  note,  it  was  observed  that  a  certain  part  of  the 
upper  chord  made  a  skip  of  a  third : 


Ex.  120. 


From  this  may  be  deduced  the  following  conclusions  : 

1.  The  first  note  of  the  skip  is  the  fifth  of  the  first  chord,  and 
the  second  note  of  the  skip  becomes  the  root-note  (or  octave)  of  the 
second  chord. 

2.  In  a  descending  skip  the  above  order  is  reversed;  z.  e.y  the 
root-note  of  the  first  chord  (in  the  melody)  descends  a  third  to  the 
fifth  of  the  second  chord. 

3.  In  an  ascending  skip  the  base  part  descends  a  second ;  in  a 
descending  skip  it  moves  up  a  second. 

Therefore  the  two  methods  of  harmonizing  a  skip  of  a  third  are 
as  follows : 

1.  With  any  chord  that  contains  both  notes  of  the  skip. 

2.  With  two  different  chords,  containing  no  connecting  note 
between  them.     Put  into  practical  operation  the  latter  method  first : 


Ex.  121. 


The  first  note  is  to  be  considered  as  fifth  of  the  first  chord ;  the 
second  note  of  the  skip  is  the  octave  of  the  second  chord.  Henco 
the  figures  above  the  melodic  notes. 

(Write  the  chords  beneath  these  notes.)     *     *     * 


GOODRICH'S  ANALYTICAL  HARMONY. 


6t 


If  c  is  the  fifth  of  a  chord,  F  must  be  the  root,  five  degrees 
below.  The  eighth  or  fifteenth  degree  below  £  is  the  root  of  the 
second  chord.  See  complete  example  : 


Ex.   122. 


This  progression  was  mads  in  the  twentieth  combination  of  the 
thirty  harmonic  progressions.  But  in  that  case  it  was  written 
harmonically,  from  the  base  part ;  here  it  occurs  melodically ;  the 
soprano  part  having  the  skip. 

By  reversing  the  above  progression,  the  harmonization  of  the 
descending  skip  of  a  third  results : 


Ex.  123. 


The  figures  and  movement  of  the  parts  are  here  directly  reversed. 
(Compare  Exs.  122  and  123.)  Both  these  melodic  skips  might  be 
harmonized  with  the  A-minor  or  with  the  C-major  triads.  For  the 
present,  harmonize  these  skips  with  unconnected  chords  as  indicated 
by  Exs.  122  and  123: 
Ex.  124. 


E E — «_ 


The  figures  refer  to  the  roots  of  the  chords  below.  3  signifies  that 
the  melodic  note  is  third  of  the  accompanying  chord.  The  example 
is  to  be  completed  in  four  parts.  *  *  * 

When  the  theme  skips  a  third  there  is  (in  this  example)  no  con- 
necting note  between  the  two  upper  chords.  The  examples  should 
be  examined  attentively,  and  every  peculiarity  of  this  kind  observed. 

Transpose  the  theme  into  D  and  B-flat  major,  and  harmonize 
accordingly.  *  *  * 

The  next  is  a  more  complete  example,  and  is  to  be  harmonized 
without  connecting  notes,  excepting  where  the  melody  moves  alpha- 
betically : 


62                      GOODRICH'S  ANALYTICAL  HARMONY. 

n     i                                                       .-r,                ss      .->     "*"     .  &      ,       "*~ 

_             CjCdZ3C2ZZZ2 

1       in       £ 

(?       | 

i           !                    1 

Ex.  izg.LJL-b-'.'* 

(? 

1                               I 

H« — F-'S^ 


5    ditto. 


The  last  measure  is  to  consist  of  the  tonic  harmony. 

Transpose  the  melody  into  D-flat  and  B  and  harmonize  similarly. 

*fC          5fC         Sji 

•  There  are  now  two  methods  for  harmonizing  melodic  skips  of  a 
third.  It  will  be  well  to  put  these  into  practice  before  concluding  this 
chapter.  Which  ever  of  these  methods  is  employed  in  a  certain  pa.'v- 
sage  will  depend  upon  the  nature  of  the  sentiment,  or  the  fancy  of 
the  composer.  For  instance,  the  notes  c  and  e  may  be  accompanied 
in  three  different  ways  : 


Ex.  126. 


The  first  arrangement  (a)  is  bright ;  (b)  is  rather  sombre ;  (c)  is  dis- 
connected and  somewhat  bold.     All  are  correct. 

Transpose  this  example  into  B-flat  and  D.     The  following  motive 
may  be  harmonized  in  different  ways : 


Ex.  127. 


Examine  and  transpose  the  example.     *    *     * 

The  skip  of  a  fourth  can  be  harmonized  in  but  one  way :  \vitri 
that  chord  which  contains  both  notes  of  the  skip,  as  explained. 

A  theme  will  now  be  presented  in  which  the  skips  of  a  third  are 
to  be  harmonized  according  to  the  different  methods  already  set  forth : 


GOODRICH  S    ANALYTICAL    HARMONY. 


£x.  128. 


Pursue,  as  much  as  possible,  the  plan  adopted  in  Ex.  127. 

The  theory  thus  far  developed  embraces  the  whole  modus  operandi 
of  handling  chords.  From  these  principles  there  will  be  no  deviation 
before  reaching  the  chapter  entitled  Unrulable  Progressions.  Even 
these,  however,  will  not  detract  from  the  value  of  the  theory  of  chord 
progression  and  harmonization. 

Transpose  the  last  example  into  various  keys. 


GOODRICH  s  ANALYTICAL  HARMON  y. 


PART  IV. 


Chapter  XVJ, 


THE  HARMONIC  MINOR  SCALE  AND  ITS 
CONSONANT  TRIADS. 

The  minor  scale  was  probably  so  called  on  account  of  the  'thirc 
and  sixth,  which  intervals  are  smaller  than  in  the  major. 

There  are  several  species  of  minor  scale,  all  serving  a  purpose  ir 
musical  composition.  Only  the  harmonic  form  will  now  be  examined 

Modern  tonality  requires  thai  the  seventh  of  every  normal  scale 
shall  be  a  minor  2d  below  the  t^nic,  as  a  leading  tone.  Beginning 
upon  A,  the  result  is  as  follows : 

EX. 


The  chief  peculiarities  of  this  scale  are  :  three  half  steps  and  one  step 
of  an  augmented  2d,  6  to  7.*  This  is  true  so  long  as  we  remaii: 
exclusively  in  the  minor  key. 

An  elementary  view  of  the  harmonic  possibilities  of  this  scale 
will  now  be  given. 

Write  a  triad  upon  each  note  of  the  scale,  being  careful  to  use 
only  the  notes  of  the  scale  already  quoted.  Then  examine  each  triad 
in  order  to  determine  the  concords.  (A  consonant  triad  must  con 
tain  a  normal  5th  and  a  major  or  a  minor  3d.) 

Every  triad  that  is  not  consonant  should  be  excised,  leaving  the 
concords.  *  *  * 

The  second  and  seventh  triads  are  imperfect  and  the  third  is  aug- 
mented. An  example  of  this  is  presented  for  comparison  : 

*Any  maior  or  normal  interval  becomes  "augmented"  when  enlarged  by  achromatic  step. 


UUODRICH  S    ANALYTICAL    HARMONY. 


Discords  excised. 


Ex.  130. 


Concords  remaining. 


These  concords  are  composed  of  the  natural  notes  of  the  harmonic 
minor  scale.  Observe  that  g-natural  does  not  appear.  These  four 
concords  are  known  by  the  following  names : 

i.     Chord  of  the  Tonic,  founded  upon  the  key-tone. 

4.  Chord  of  the  Sub-dominant,  founded  upon  the  4th  above,  or 
the  5th  below  the  tonic. 

5.  Chord  of  the  Dominant,  so  called  because  it  dominates  or  con- 
trols the  key.     In  major  and  minor  this  chord  is  founded  upon  the 
fifth  natural  degree. 

6.  Chord  of  the  Sub-mediant. 

From  these  four  concords  a  chart  is  to  be  made,  showing  how- 
many  chords  will  accompany  each  note  of  the  scale. 

Note.  In  founding  chords,  write  a  3d  and  5th  above  the  foundation-note, 
or  root,  but  in  preparing  a  chart  or  harmonizing  a  theme,  the  chords  are  writ- 
ten below  the  melodic  notes.  *  *  * 

This  diagram  is  presented  for  comparison : 


Do 


A  short  theme  can  now  be  harmonized,  exclusively  in  this  minor 
scale.  The  fact  that  three  notes  of  the  scale  can  be  accompanied 
with  only  one  chord  each,  leaves  us  no  choice  in  the  harmonization 
of  those  degrees.  But  as  we  wish  to  remain  entirely  in  this  scale, 
and  have  nothing  but  concords  to  work  with,  we  must  be  governed 
by  the  chart : 

THEME   IN   A-MINOR. 
Do      si      do       do      re     mi        fa      mi     re         do     do      si        do. 

^  n   . 
Ex.  132. 

The  letters  above  the  melody  correspond  to  those  of  the  chart,  and 
render  the  harmonization  easy  of  accomplishment.  The  repeated 
notes  are  to  be  considered  connecting  links,  and  the  harmony  is  to 
change  at  such  places. 

Like  previous  themes,  this  is  continuous,  and  should  begin  and 
end  with  the  tonic  chord. 


66 


GOODRICH'S  ANALYTICAL  HARMONY. 


The  chord  progressions  must  in  every  instance  be  according  to 
previous  directions.  The  interval  of  an  augmented  2d  (/to  g-sharp) 
appears  in  the  second  measure  ascending,  and  from  the  third  to  the 
fourth  measures  descending.  The  author  regrets  that  this  interval 
has  been  forbidden  by  theorists,  for  if  it  is  incorrect,  then  the  scale 
must  be  incorrect.  Nothing  but  a  spirit  of  mechanical  and  anti- 
artistic  pedantry  could  have  sought  to  interdict  an  interval  so  neces- 
sary to  composition,  and  so  characteristic  of  our  modern  minor  scale. 
Do  not,  therefore,  hesitate  to  use  the  augmented  2d ;  it  is  perfectly 
proper,  and,  in  fact,  indispensable.* 

Another  theme  is  offered,  affording  more  scope  in  the  harmoniza- 
tion : 

Ex.  133. 


& 


f2—^ 


This  is  exclusively  in  G-minor. 

As  it  presents  no  new  difficulties,  its  completion  is  left  to  the 
student,  with  the  advice  that  it  be  arranged  carefully  and  transposed 
into  other  minor  scales. 


Chapter  XVII. 


THE  MAJOR  AND  MINOR  MODES  COMBINED. 

INTRODUCTION  TO  MODULATION. 

WHILE  we  remain  strictly  in  a  minor  key  it  is  evident  that  our 
means  are  limited,  and  our  field  of  operations  narrow. 
The  four  concords  can  be  combined  into  twelve  progressions ; 
still  we  feel  restrained,  having  but  one  chord  with  which  to  accom- 
pany the  2d  and  yth,  and  but  one  for  the  4th  of  the  scale.     We  will, 
therefore,  combine  the  relative  major  and  minor  modes,  as  they  are 
intimately  related  and  connected. 


*This  applies  to  instrumental  music.    In  vocal  compositions  the  augmented  ad  is  fre- 
quently undesirable. 


GOODRICH'S  ANALYTIC^:,  HARMONY. 


67 


The  minor  scale  constitutes  a  mode,  or  characteristic  series  of 
sounds  having  a  recognizable  fundamental,  or  key-tone,  to  which  the 
melodic  or  harmonic  cadence  naturally  resolves. 

The  major  scale  constitutes  another  mode,  possessing  equally 
recognizable,  though  different  characteristics. 

It  is  proposed  to  combine  these  two  modes. 

Reference  is  here  made  to  relative  major  and  relative  minor, 
having  the  same  external  signature.  Each  scale  contains  a  tone 
foreign  to  the  other.  A-minor  includes  g-sharp  as  leading  tone, 
while  C-major  embraces  g-natural  as  normal  5th  and  foundation  of 
its  dominating  harmony.  Thus  each  scale  has  an  individual  tone 
not  contained  in  the  other.  From  this  it  is  seen  that  g-sharp 
points  to  A-minor  and  g-natural  to  C-major. 

In  combining  the  chords  for  this  dual  key,  three  of  the  chords  in 
A-minor  already  occur  in  C-major.  Name  the  chord  in  A-minor  that 
does  not  appear  among  the  six  in  C-major. 

With  the  addition  of  this,  there  will  be  seven  triads  to  work  with. 
Concords  occurring  in  both  scales  are  given : 


Ex.  134. 


The  upper  figures  refer  to  the  major,  the  lower  figures  to  the  minor 
scale. 

Write  a  chart  for  both  modes,  beginning  upon-C.  In  adding  the 
chords  beneath  the  scale  degress  the  chord  containing  g-sharp  is  to 
be  included  every  time  one  of  its  notes  occur  in  the  upper  part. 

This  dominant  chord  to  the  relative  minor  is  to  be  written  last, 
for  it  is  not  well  to  follow  the  chord  containing  g-sharp  with  a  chord 
containing  g-natural.  The  indication  of  the  chart  is  to  be  completed 
by  the  student : 

Ex.  135. 


The  theory  for  applying  this  chart  to  harmonization  has  already  been 
partially  explained. 

When  g-natural  appears,  either  melodically  or  harmonically,  the 
tonality  of  c-major  prevails.  But  when  g-sharp  makes  its  appear- 
ance, the  minor  key  is  anticipated,  for  it  is  leading-note  to  A-minor, 


68 


GOODRICH'S  ANALYTICAL  HARMONY. 


and  3d  of  the  dominant  harmony.     The  minor  mode  will  prevail 
until  one  of  these  chords  occur : 


Consequently  the  chords  E-major  and  E-minor  should  be  considered 
entirely  distinct  from  each  other.  In  connection  with  this  subject 
the  author  will  promulgate  but  one  arbitrary  rule,  and  this  must 
remain  in  force  until  more  experience  on  the  part  of  the  student 
shall  justify  its  violation  :  Whenever  use  is  made  of  the  dominant 
chord  to  the  related  minor,  it  must  be  followed  by  one  of  these 
chords :  Tonic-minor,  sub-dominant  or  sub-mediant.  In  other 
words,  g-sharp  must  not  be  followed  by  g-natural.  No  objection 
can  be  made  to  the  reverse  of  this  order.  (See  chart.) 

The  theme  which  will  presently  be  introduced  is  so  contrived  as 
to  be  alternately  in  C-major  and  A-minor.  In  this  respect  it  is  similar 
to  many  compositions  of  the  present  day.  The  tonality  is  generally 
influenced  by  the  harmony.  For  instance,  this  melodic  phrase  might 
be  harmonized  in  three  or  four  different  ways : 


Ex.  137. 


i=± 


(It  would  be  well  for  the  student  to  attempt  several  arrangements 
of  this  phrase.)     *     *    * 

It  may  begin  and  end  in  either  C-major  or  A-minor.     As  a  prep- 
aration to  what  follows,  these  examples  are  presented : 


Ex.  138. 


•EJ  fr  |  j  r  r  ^E| 


All  these  harmonizations  are  correct,  though  each  has  its  peculiar 
effect  and  distinct  application  in  actual  composition.  But  the 
esthetic  character  of  the  examples  can  not  be  considered  here. 

EX.  139.  THEME  UN  C-MAJOR  AND  A-MINOR. 


Begin  in  C-major. 


Temporarily  in  A-minor. 


End  in  C-major. 


GOODRICH'S  ANALYTICAL  HARMONY. 


69 


This  melody  is  principally  in   C-major.     In  the  fourth  measure 

there  is  a  transient  passage  to  the  relative  minor. 

Harmonize  as  usual  in  four  parts,  and  then  transpose.     *    *    * 
The  next  theme  is  to  be  harmonized  principally  in  A-minor.     In 

the  fifth  measure  there  is  a  temporary  digression  to  the  relative 

major : 

Ex.  140. 


The  g-sharp  is  the  only  note  that  admits  no  choice  of  harmony. 
Transpose  as  in  the  last  example.     *    *     * 

One  more  theme  is  offered  for  harmonization  according  to  the 
same  chart.     This  should  begin  in  C  and  end  in  A-minor  : 
Ex.  141. 


£ 


r- 


£ 


1 


The  main  requisites  are  to'  have  the  chord  progressions  correct, 
and  to  know  where  the  changes  of  mode  occur,  as  this  lesson  is  in- 
tended to  foreshadow  the  principles  of  transition. 

Transpose  each  example  until  the  subject  it  illustrates  is  thor- 
ougly  comprehended. 


Chapter  XVIII. 


PRIMARY  MODULATIONS  TO  ALL  THE   RELATED 
KEYS  EXCEPTING  THE  SUB-DOMINANT. 

MODULATION  signifies  a  change  of  key,  a  passage  to  some 
other  base  of  operations.     It  is  generally  used  synonymously 
with  Transition,  though  the  latter  is  the  stronger  term.     This  dis- 
tinction will  appear  hereafter. 

To  accomplish  even  a  temporary  modulation  some  chromatic 
alteration  must  be  introduced  that  is  suggestive  of  the  key  to  be 
established.  For  instance,  in  modulating  from  C-major  to  the 


70  GOODRICH'S  ANALYTICAL  HARMONY. 

relative  minor  it  will  be  necessary  to  introduce  into  the  modulating 
harmony  some  note  that  is  characteristic  of  A-minor,  and  does  not 
belong  naturally  to  the  original  scale,  C. 

What  is  this  note  ?  The  dominant  to  A  is  e,  and  as  the  dom- 
inant chord  is  supposed  to  contain  a  major  3d  and  normal  5th,  the 
characteristic  note  is  found  in  this  chord : 


Ex.  142. 


This  leads  naturally  to  the  chord  of  A-minor.  The  dominant  chord 
is  best  adapted  to  perform  a  natural  modulation,  and  it  is  the  only 
modulatory  chord  at  present  available  for  our  purpose. 

A  dominant  chord  is  founded  upon  the  fifth  tone  of  any 
major  or  minor  scale,  and  contains  a  large  3d  and  a  normal 
5th  from  the  root. 

The  root  is  a  dominating  note,  and  when  it  appears  in  the  base, 
that  part  ascends  a  fourth  or  descends  a  fifth  to  the  key-tone  with 
considerable  strength  and  determination,  thus : 


Ex.  143. 


The  other  parts  of  the  chord  correspond  to  this.  The  root  remains 
as  fifth  of  the  tonic  chord ;  the  third  ascends  a  half  step  to  the  key- 
tone  ;  and  the  fifth  ascends  to  the  third  of  the  tonic  chord  :* 


Ex.  144. 


The  3d,  being  the  leading-tone,  is  the  most  important  note  of  the 
dominant  chord,  particularly  in  modulation.  The  dominant  chord 
is  the  same  in  tonic  major  or  tonic  minor.  See  example : 


-S 

A 

'   J  w  ' 

* 

G-mAjor. 


G-minor. 


EJUI45. 


'•'When  the  third  is  said  to  ascend  it  is  understood  to  refer  to  the  voice  or  instrument 
that  sounds  this  tone,  for,  strictly  speaking,  the  tone  itself  is  an  independent  sound  and  can 
uot  move. 


GOODRICH 'S   ANALYTICAL   HARMONY. 


It  is  somewhat  singular  that  this  first  chord  should  occur  naturally 
in  both  modes,  presupposing  that  the  /-sharp,  as  leading-note,  occurs 
in  the  minor  scale.  There  is  no  difference  between  the  dominant 
chord  at  (a)  and  the  one  at  (b),  though  the  first  leads  to  G-major 
and  the  second  to  tonic  minor.  This  is  an  Authentic,  or  regular 
Cadence,  and  in  future  examples  will  play  a  more  important  part. 

The  first  modulations  are  to  the  related  keys.  The  related  keys 
are  "those  which  differ  from  the  original  by  not  more  than  one  sharp 
or  one  flat."  Therefore  the  related  keys  to  C-major  are,  F-major,  G- 
major,  the  related  minors  to  these  (having  the  same  signatures)  and 
the  relative  minor  to  the  tonic.  This  family  group  consists  of  six 
keys,  three  major  and  three  minor.  The  six  concords  that  have  been 
used  are  the  tonic  triads  of  these  related  keys.  But  when  the  key  of 
any  of  these  is  mentioned,  the  full  scale,  and,  consequently,  the  signa- 
ture of  that  key,  must  be  comprehended. 

The  chords  of  these  related  keys  are  these  : 


Ex.  146. 


With  their  respective  signatures  they  would  appear  like  this : 
c        A  F        D  G        E. 

EX.  147-  E3E 


In  each  exercise  the  relative  minor  follows  its  major  with  the  same 
signature.  Therefore  when  we  think  of  a  modulation,  for  e.  g.,  to 
E-minor,  we  must  also  think  of  this  scale : 


Ex.  148. 


for  in  order  to  establish  the  tonality  of  E-minor  it  is  necessary  to 
comprehend  this  series  of  tones. 

The  technical  names  of  the  related  keys  and  the  chords  which 
represent  them  were  explained  in  Chapters  III  and  IV. 

Ex.  147  shows  the  groups  of  natural  chords,  with  their  corre- 
sponding numbers,  and  their  connection  with  each  other  harmonic- 
ally. We  may  move  the  parts  from  one  chord  to  any  other  chord 
in  the  scale  without  affecting  the  original  key-tone,  and  without 
taking  cognizance  of  any  other  scale.  This  would  be  Progression. 


72  GOODRICH'S  ANALYTICAL  HARMONY. 

But  in  order  to  effect  a  transition  and  create  a  new  tonality  we  must 
introduce  that  tone  which  represents  the  difference  between  the 
original  key  and  the  one  to  which  we  are  in  transit.*  In  this  in- 
stance the  old  key  disappears  (even  though  but  temporarily),  and  a 
new  base  of  tonal  operations  is  established.  In  the  latter  instance 
a  chromatic  alteration  is  employed  as  transition  note ;  in  the  former, 
no  chromatic  sign  is  used. 

All  the  related  chords  in  progression  follow  : 


Ex.  149. 


This  is  a  mere  chord  succession ;  the  tonality  of  C-major  not  being 
in  any  way  affected.  But  when  we  consider  any  of  these  chords  as 
tonic  of  a  related  key,  it  implies  that  a  modulation  has  been,  or  is  to 
be  accomplished. 

To  proceed  with  the  modulations  in  regular  order :  C  to  D-minor  ; 
C  to  -E-minor  ;  (C  to  F-major  here  omitted)  C  to  G-major  and  C  to 
A-minor. 

The  mode  of  each  related  key  is  governed  by  the  signature  of  the 
original  key.  In  other  words,  the  tonic  chords  to  the  related  keys 
are  to  remain  as  we  find  them  naturally : 


Ex.  150 


Mi 


Ma 


Ma 


Mi. 


If  we  should  modulate  from  C  to  D-major,  it  would  not  be  an  ele- 
mentary modulation,  but  a  transition. 

The  modulations  should  be  written  in  this  order  :  Begin  with  the 
C  chord,  c  being  uppermost.  The  first  passage  is  to  D-minor.  What 
is  the  dominant  to  D?  Write  the  root  in  the  base.  What  is  the 
leading  tone  to  D  ?  (It  must  be  a  minor  2d  below  the  key-tone.) 
What  is  the  dominant  chord  to  D  ?  Does  it  contain  the  new  leading- 
tone?  (Every  dominant  chord  must  have  a  major  3d  and  normal 
5th.)  Write  this  chord  over  the  root  note.  The  chord  movements 
must  be  correct  according  to  previous  directions.  Therefore  the 


*By  means  of  a  dominant  7th  chord  with  its  3d  omitted  we  might  prove  an  exception 
to  this,  but  it  would  be  premature  here. 


GOODRICH'S  ANALYTICAL  HARMONY.  73 

connecting  notes  must  remain  in  the  same  parts  as  usual.  Resolve 
the  second  chord  (the  modulatory  one)  to  the  chord  of  the  new  key, 
merely  following  the  rules  of  progression  in  passing  from  chord  to 
chord.  *  *  * 

Supposing  this  much  to  have  been  written  by  the  student,  the 
example  may  be  compared  to  the  following,  in  order  to  correct  pos- 
sible errors,  or  to  confirm  impressions  rightly  formed : 

i        i 

•  ^j     ^     » 
Ex.  151. 


This  first  example  is  to  be  arranged  in  two  other  positions.  The 
base  will  remain  the  same  during  the  re-arrangements  of  any  particu- 
lar modulation.  In  this  example  the  chromatic  note,  c-sharp,  serves 
to  erase  our  impression  of  the  original  key,  and,  together  with  the 
dominant  chord  on  A,  establishes  the  key  of  D-minor. 

Considering  this  as  an  individual  example,  we  begin  again  with 
the  C  chord  and  proceed  to  modulate  to  the  next  key  in  regular  order. 
What  is  the  dominant  to  E  f  Write  it  in  the  base.  What  is  the  full 
dominant  chord  ?  Does  it  contain  the  leading  tone  to  E?  Does  it 
contain  any  other  tone  not  common  to  C?  Are  these  chromatically 
altered  notes  common  to  the  scale  of  E-minor  ?  Write  the  scale  to 
prove  this  : 


Ex.  152. 

J 


The  crosses  show  the  notes  that  comprise  the  dominant  chord  to 
E-minor,  Therefore  the  chord  B,  d-sharp,f-sharp,  is  perfectly  nat- 
ural to  this  key. 

Now  write  the  modulatory  chord.  In  doing  so,  remember  there 
is  no  connecting  note  between  the  first  two  chords,  and  proceed  ac- 
cordingly. 

The  E-minor  chord  naturally  follows  the  dominant  chord  on  B, 
and  this  progression  is  easily  written.  There  is  always  a  note  of 
connection  between  dominant  and  tonic  chords;  tonic,  in  this  in- 
stance, referring  to  the  new  key-tone. 

This  modulation  to^is  to  be  numbered  2,  and  re-arranged  in  two 
other  positions.  *  *  * 


74  GOODRICH'S  ANALYTICAL  HARMONY. 

The  next  modulation  is  to  F.  The  dominant  chord  is  C,  e,  g,  but 
it  contains  no  note  foreign  to  the  key  of  C,  and  therefore  will  not 
perform  the  transition : 


Ex.  153. 


This  is  a  mere  progression  from  the  C  to  the  F  chords,  and  the  key 
still  remains  C.  Consequently  this  modulation  to  the  sub-dominant 
must  be  omitted,  as  it  can  not  be  accomplished  without  a  discord. 

The  key  of  G  is  next  to  be  established.  The  same  theory  will 
serve  our  purpose.  What  is  the  dominant  to  G  ?  (Always  the  5th 
of  any  scale.)  What  is  the  dominant  chord  to  G?  Does -it  contain 
the  leading-note  to  G  ? 

Write  the  base  first  after  the  chord  of  C,  with  c  uppermost,  add 
the  dominant  chord  above  the  root,  and  end  with  the  chord  of  the 
new  tonic  G. 

(Between  the  C  and  D-major  chords  there  is  no  connection.  The 
upper  parts  must  accordingly  move  in  an  opposite  direction  to  that 
of  the  base.)  Two  other  positions  of  this  example  are  to  be  written, 
as  usual.  In  going  from  dominant  to  tonic,  the  fundamental  base 
may  ascend  a  4th  or  descend  a  5th  to  the  key-tone. 

The  fourth  of  the  present  modulations  is  to  the  relative  minor. 
Write  the  base  first,  then  the  other  parts,  being  careful  to  use  the 
major  3d  of  the  dominant  chord  as  leading  tone  to  the  new  key. 
Number  this  4,  and  re-arrange  in  two  other  positions. 

When  all  these  are  completed,  write  the  same  modulations  from 
D-major,  E-major  and  B- flat-major. 

In   no   instance   is  the   modulation   to   the  sub-dominant   to   be 
attempted. 


GOODRICH'S  ANALYTICAL  HARMONY. 


Chapter  XIX. 


THEMES    FOR    HARMONIZATION,    ILLUSTRATING 
THE   PRECEDING   MODULATIONS. 


HE  separate  modulations  which  have  been  accomplished  from  an 
•»•   original  key-tone  will  now  be  included  in  a  continuous,  transi- 
tional melody. 

The  student  is  first  to  discover  what  modulations  are  intended  at 
certain  points  in  this  theme,  and  then  how  these  modulations  are  to 
be  made.  (The  latter  problem  has  been  solved.) 


MODULATORY   THEME. 


The  chromatic  notes,  even  without  the  dashes,  would  indicate 
where  the  modulations  take  place.  By  referring  to  the  Table  of 
Modulations  in  B-flat,  something  of  a  chart  will  be  found  for  the 
harmonization  of  this  modulatory  theme.  The  only  difficulty  is  that 
this  melody  is  continuous.  The  modulations  do  not  begin  with  the 
chord  of  the  original  key-tone,  as  was  the  case  in  the  mechanical  ex- 
amples. For  instance,  the  first  chord  in  the  third  measure  is  not 
B-flat,  but  either  G-minor  or  D -minor.  The  chord  movements 
must,  however,  be  written  correctly. 

The  e-natural,  in  the  third  measure,  might  be  mistaken  for  a 
modulatory  indication  other  than  that  intended.  But  the  following 
c-sharp  removes  the  doubt,  as  it  is  not  intended  to  modulate  to  the 
same  key  in  two  different  places. 

After  performing  the  modulation  to  C-minor  it  would  be  well  to 
introduce  the  G-minor  chord  on  the  first  of  the  third  measure  in 
order  to  temporarily  restore  the  original  tonality ;  for  it  is  more  ele- 
mentary to  modulate  from  G-minor  to  F-major  than  from  C-minor  to 
F-major.  In  the  last  of  the  fifth  measure  (third  quarter)  the  impres- 
sion of  the  modulation  to  D-minor  is  to  be  erased  by  using  the  domi- 
nant chord  to  the  original  key.  After  the  modulation  to  G-minor, 


GOODRICH'S  ANALYTICAL  HARMONY. 


include  the  dominant  chord  to  B-flat  in  order  to  restore  the  original 
tonality. 

In  case  any  student  should  experience  difficulty  with  this  example 
it  may  first  be  transposed  into  C. 

When  the  example  is  completed,  the  mezzo-soprano  and  con- 
tralto parts  should  each  be  taken  as  a  theme,  including  the  previous 
modulations.  This  will  merely  result  in  two  other  arrangements  of 
the  same  modulations,  but  it  will  also  show  how  modulations  may 
be  effected  without  any  outward  sign,  thus  illustrating  the  harmonic 
possibilities  of  a  melody.* 

These  resultant  themes  have  been  extracted  from  the  original 
harmonization,  and  are  here  presented  for  the  student's  benefit.  As 
the  same  harmonies  and  modulations  are  to  apply  to  these  additional 
arrangements  the  base  will  require  no  alteration  ; 


Ex.  i 

2 

55 

Mezzo-soprano  part  as  theme. 

ZBB$ 

gS 

3 

•••      d  * 

<eC     W  i 

^        f» 

*    j   y 

-*-  f 

II  I  '  — 

V  1' 

E_ 

&-       9 

I 

*  * 

c  . 

.  J  *    i& 

3. 

Contralto  part  an  8va  higher  as  theme. 

—j  -  ?  i  &  0    i  r    »Vu  »  i  »  1  —  I  1—  i  —          —  .  • 

~»r>b 

'  '  •  [-"""• 

19       F   | 

1  

H  »t*- 

CpC 

w  1  — 

^=»3=E- 

h^rv}r 

—  I  —  |  —  i 

—  i-^H 

Sf.  — 

1  1 

Transpose  these  themes  into  C  and  . 

D-flat,  and  harmonize.     For  this 

purpose  a  chart  may  be  necessary. 

The  next  theme  contains  no  chromatic  notes  to  indicate  the 
modulations,  but  a  reference  to  the  chart  in  C  will  show  the  solu- 
tion. Two  d's  for  instance  will  be  found  in  the  transition  to  G-major. 
In  the  original  table  the  two  d's  appear  in  the  second  arrangement. 
This  will  serve  as  a  clue  to  the  others. 


THEME. 


Ex.  156. 


Harmonize  and  transpose  into  A,  B,  and  D.~\ 

The  chords  should  also  be  re-arranged  if  the  pupil  needs  still 
more  practice  in  this  subject. 

*  A  minor  key  may  be  established  by  means  of  minor  chords  without  employing  a  tran- 
sition chord.  The  same  may  be  done  with  a  major  key.  But  this  can  not  be  explained  at 
the  present  time. 

t  Remember  that  the  J3  is  a  sign  of  elevation  in  a  flat  Ley,  and  of  depression  in  a  sharp 
key. 


GOODRICH'S  ANALYTICAL  HARMONY. 


Chapter  XX. 


MODULATIONS   FROM  THE   MINOR  MODE,  WITH 
ILLUSTRATIVE  THEMES. 

TN  making  modulations  from  a  minor  key- tone  there  are  certain 
-A-  principles  to  be  understood  that  have  merely  been  touched  upon 
in  previous  chapters.  These  principles  will  be  explained  as  they 
occur  in  the  following  modulations.  The  intention  is  to  perform 
modulations  from  a  minor  key-tone,  not  to  make  mere  progressions. 
The  following  triads  are  selected : 


The  upper  figures  indicate  the  order  in  which  the  keys  are  classified, 
first  minor  and  then  major ;  the  lower  figures  show  the  degrees  of 
the  scale  upon  which  these  triads  are  founded. 

Observe  that  the  dominant  chord  here  appears  as  minor,  because 
a  natural  modulation  can  not  be  made  to  E-major. 

But  as  the  signature  of  E-minor  is  only  one  sharp  it  is  included 
among  the  related  keys.  The  leading-note  to  A-minor  does  not  ap- 
pear, because  the  object  is  to  show  the  keys  to  which  modulations 
are  to  be  made.  The  last  chord  ((7)  is  founded  upon  the  subtonic, 
or  imperfect  leading-tone. 

Begin  with  the  A-minor  chord  (as  representing  the  key  of  A- 
minor}  and  modulate  to  D-minor.  This  is  to  be  done  in  the  same 
manner — by  means  of  the  dominant  major  chord.  (The  transition 
chord  is  the  same  as  was  used  in  going  from  C  to  D-minor.} 

Then  begin  again  in  A-minor  and  modulate  to  E-minor,  as  before. 
The  modulation  to  C-major  can  be  effected  in  the  same  way ;  but  it 
must  be  understood  that  g-sharp  is  the  only  characteristic  tone  in  A- 
uiinor  that  does  not  occur  in  C-major.  In  commencing  in  A-minor 
and  introducing  the  dominant  chord  to  C,  which  is  founded  upon 
G-natural,  the  latter  note  acts  as  a  chromatic  alteration  and  effects 
the  change  of  key.  According  to  the  same  principle  a  passage  can 


78  GOODRICH'S  ANALYTICAL  HARMONY. 

be  made  to  modulate  from  A-minor  to  the  relative  major  of  the  sub- 
dominant,  which  would  have  been  impossible  in  C-major  without 
employing  a  discord.  For  instance  : 


Ex.  158. 


A-min.  to  F-major. 

The  chromatic  sign  in  the  second  chord  complies  with  the  principle 
at  first  set  down,  that  every  transition  chord  must  contain  a  chromatic 
alteration.  The  g-natural  does  not  belong  to  the  scale  of  A. 

The  modulation  to  the  relative  major  of  the  dominant  is  effected 
by  the  dominant  major  chord  containing  f -sharp. 

These  modulations  should  all  be  written  in  three  positions.  A 
theme  will  now  be  presented,  in  the  harmonization  of  which  the 
student  is  to  begin  in  A-minor,  modulate  to  five  related  keys,  and 
back  to  the  original  tonic  in  the  final  cadence. 

Ex.  159. 


^B  —  *  —  i  —  i  —  ' 

1     1 

i  '  ' 

-r-rH 

By  referring  to  the  separate  modulations  from  A-minor,  the  stu- 
dent will  discover  what  keys  are  to  be  established  at  the  places 
indicated  by  dashes. 

After  harmonizing  this,  the  mezzo-soprano  and  contralto  parts 
may  be  taken  as  themes  (uppermost)  and  harmonized  in  the  same 
way. 

Also  write  a  table  of  modulations  from  E  and  other  minor  keys  to 
their  relatives. 


GOODRICH'S  ANALYTICAL  HARMONY. 


PART  V. 


Chapter  XXI. 


FORMATION  AND  RESOLUTION  OF  THE 
DOMINANT  SEVENTH  CHORD. 

"YT  TE  have  arrived  at  a  point  beyond  which  no  material  progress 
*  *  can  be  made  with  concords.  A  discord  is  therefore  introduced. 
In  modulations  from  a  major  key  it  will  be  remembered  that  the  sub- 
dominant  was  omitted,  because  the  tonic  chord  could  not  appear 
simultaneously  as  tonic  and  dominant.  So  the  inquiry  is  made — 
what  note  in  the  F  scale  does  not  appear  in  the  C  scale  ?  This  note 
is  situated  a  minor  third  above  the  5th  of  the  concord,  with  which  it 
may  well  be  combined. 

The  chord  on  C  should  not  be  rejected  because  it  is  insufficient 
lo  perform  the  modulation  to  F,  but  a  transition  element  should  be 
i:dded  to  it : 


Ex.  160. 


This  destroys  the  impression  of  the  key  of  C  and  creates  the  key 
F,  because  these  notes  occur  naturally  in  no  other  scale : 


Ex.  161. 


1 


The  crosses  indicate  the  dominant  seventh  chord  just  formed. 

This  is  a  four-toned  chord,  the  most  agreeable  and  one  of  the 
most  important  in  our  harmonic  vocabulary.     It  is  called  a  discord, 


>fo  GOODRICH  S   ANALYTICAL    HARMONY. 

merely  in  opposition  to  concord,  for  the  root  and  yth  form  a  dissc 
nant  interval  and  require  resolution  to  a  consonance : 


Ex.  162. 


This  is  still  more  noticeable  when  inverted : 


To  analyze  it  farther,  it  contains,  theoretically,  a  major  3d,  norma 
5th  and  minor  yth  : 


Ex. 


It  may  also  be  described  as  consisting  of.  one  major  and  two  mine 
thirds.     Or,  it  includes  a  normal  and  an  imperfect  5th  combined . 


Ex.  165. 


In  certain  resolutions  it  will  be  necessary  to  view  it  in  this  light. 

THE  RESOLUTION. 

As  already  observed,  this  discord  belongs  to  the  key  of  F,  Th 
3d  is  leading  note,  and  ascends  a  minor  2d  to  the  tonic ;  the  5th  \\i. 
no  fixed  resolution,  but  for  the  present  will  be  directed  downwar 
a  whole  step;  the  yth  has  a  decided  natural  tendency  to  resolv 
down  to  the  3d  of  the  tonic  chord ;  the  root-note,  when  it  appear 
among  the  upper  parts,  remains  stationary,  and  becomes  5th  of  th 
tonic  concord ;  the  base  moves  from  root  to  root.  The  illustratio 
is  presented  in  score  in  order  to  show  more  plainly  the  resolution  o 
the  discord : 


GOODRICH  S   ANALYTICAL    HARMONY. 


8l 


The   most   important  notes  of  the  discord 
are  the  3d  and  yth  : 

£-a,^~r1  •"«-: 1 


Ex.  166. 


Ex- 


In    whatever   position   the   chord    may 
appear,  these  directions  will  apply  : 


These  tones  are  known  in  theory  as  "  elements  of  transition,"  and 
are  usuallj'  called  by  their  technical  terms,  "leading-tone  and  sub- 
dominant."  (These  terms  here  refer  to  the  key  of  F,  for  the  chord 
no  longer  belongs  to  C.}  Leading  tone  always  means  the  minor  2d 
below  any  tonic,  and  sub-dominant  refers  to  the  4th  of  the  key  to 
which  the  discord  belongs.  These  important  elements  will  be  met 
with  in  other  discords,  and  their  thorough  comprehension  will  facili- 
tate future  labors. 

The  dominant  yth  chord  must  now  be  written  and  resolved  in 
every  major  key  and  in  every  position.  As  the  base  merely  moves 
from  root  to  root,  the  examples  may  be  written  without  the  base 
staff,  as : 


In    C. 


EX.  i6g.  bfer: 


I   I 


As  each  note  of  the  discord  appears  lowermost  in  regular  succession, 
the  re-arrangement  of  the  discord  will  present  no  difficulty.  Observe 
that  each  note  of  the  discord  is  resolved  in  the  same  -manner  at  2,  3 
and  4,  as  at  i.  Proceed  by  fifths  in  the  transpositions  and  include 
the  signature  of  each  key.  The  root  of  the  discord  will  then  appear 
upon  the  5th  of  each  scale,  and  no  further  chromatic  sign  will  be  re- 
quired. Here  is  a  partial  indication  of  the  next  example  in  order: 

In  G. 

Ex.  170. 


As  the  3d  ascends  and  the  5th  descends  to  the  tonic,  there  are  two 


82  GOODRICH  S  ANALYTICAL  HARMONY. 

voices  singing  this  latter  tone,  and  it  is  well  to  indicate  this  fact  fo? 
the  present  by  writing  the  tonic  note  in  unisons  or  primes.  This  is 
explained  by  the  third  position  ot  each  example,  and  also  by  the  five 
part  resolution  in  score,  Ex.  166. 

In  the  first  exercises  it  is  advisable  to  resolve  the  most  important 
elements  first.  These  are  the  sub-dominant  and  leading-tone  ( jth 
and  3d  of  the  discord),  and  in  whatever  position  the  chord  may  be 
these  names  still  apply,  and  the  resolution  is  the  same. 

B,  F-sharp  and  C-sharp  should  be  written  in  their  enharmonic 
equivalents ;  i.  e.,  with  the  same  sounds,  but  different  notation.  An 
example  of  this  process  is  presented  : 


Ex.  171. 


F-sharp  becomes  g-flat,  a-sharp  becomes  b-flat,  c-sharp  becomes  d-flat, 
and  e  will  appear  as  f-flat.  (A)  and  (b)  are  enharmonic  equivalents. 
The  first  belongs  to  B-major,  the  second  to  C -flat-major.  Both  keys 
are  practically  identical.  F-sharp  and  G-flat  are  also  equivalent  keys, 
and  so  are  C-sharp  and  D-flat. 

Continue  the  transpositions  (Ex.  170)  by  fifths,  and  thus  return  to 
C.  Such  process  is  called  the  cycle  of  keys.  The  student  shou/d 
complete  the  task.  *  *  * 

If  the  base  be  included  with  the  exercise,  five-part  harmony  will 
result,  as  in  Ex.  166.  This  shows  the  resolution  of  all  the  parts  of  a 
discord ;  but  it  is  not  advisable  to  employ  more  than  four  parts  in  the 
present  exercises. 

The  remaining  modulation  omitted  in  previous  lessons  should 
now  be  supplied.  Begin  with  the  C  chord  with  c  uppermost ;  retain 
the  root,  3d  and  5th,  and  move  the  soprano  part  from  c  down  to  b-flat. 
In  the  next  measure  resolve  the  discord  according  to  previous  direc- 
tions. The  base  may  ascend  a  4th,  or  descend  a  5th,  from  root  to 
root.  *  *  * 

Write  two  re-arrangements  of  this.  An  example  of  the  three 
arrangements  is  offered  for  comparison  : 


Ex.  172. 


This  completes  the  modulations  from  C. 


GOODRICH'S  ANALYTICAL  HARMONY. 


6 

r"  Z      '          <&•«>- 

~~ 

»       1% 

W    1 

^_5!  1  

r 

?&} 

V&  H^J— 

—  —  1 

^ 

1                              2 

3 

c* 

^>  

C* 

|-»  

1 

2 





^— 

The  sub-dominant,  b-flat,  is  here  the  most  important  element  of 
transition,  as  it  represents  the  difference  between  the  scales  of  F 
and  C.  The  modulation  corresponding  to  this  in  the  key  of  G  is 
to  C-major.  It  is  to  be  accomplished  in  the  same  manner : 


Ex.  173. 


Continue  these  exercises  in  several  keys,  until  the  principle  is  well 
understood. 

This  mode  of  treatment,  though  perfectly  proper,  leaves  the  final 
chord  incomplete.  Observe  that  no  5th  appears  in  the  concord. 
The  root  and  3d,  however,  give  a  fair  representation  of  the  tonic 
chord,  for  those  notes  occur  in  but  one  other  concord,  the  relative 
minor,  and  we  are  not  inclined  to  imagine  this  latter  chord. 

The  root  in  the  base  gives  a  very  strong  indication  of  being  tonic, 
and  the  resolution  of  the  discord,  according  to  its  most  natural  ten- 
dency, serves  to  confirm  the  impression  created  by  the  base.  This, 
therefore,  may  be  considered  as  synonymous  with  the  cadence  in 
which  the  concord  is  fully  represented : 


Ex.  174, 


(A)  has  been  used  as  frequently  as  (b),  especially  on  the  final  ca- 
ieuce.     Following  is  an  instance  from  a  standard  English  Glee : 

Callcott. 


:?c=: 


Ex.  175. 


From  this  and  innumerable  similar  instances  it  is  reasonable  to  con- 
clude that  the  5th  of  the  concord  is  not  essential  in  a  final  cadence. 


84         GOODRICH'S  ANALYTICAL  HARMONY. 

But  in  an  intermediate  progression  the  incomplete  concord  is  some- 
times difficult  to  manage.     Witness  these  examples  : 


EfeSEfeEfei 


etc. 


m 


U-^-J-j-LJZZ^ 


etc. 


•£ 


The  progressions  indicated  by  the  dashes  are  awkward,  if  not 
positively  incorrect.*  To  obviate  these  difficulties  will  be  the 
principal  object  of  the  next  chapter. 


Chapter  XXII. 


OMISSION  OF  THE  THIRD  OR  THE  FIFTH   FROM 
THE  DOMINANT  SEVENTH  CHORD.f 

A  NOTHER  mode  of  treating  the  dominant  yth  chord,  so  as  to 
-^"^  leave  the  concord  complete,  will  now  be  shown.  The  root  of 
the  discord  is  the  same  as  the  5th  of  the  tonic  triad,  and  is,  there- 
fore, a  connecting  note : 


Ex.  178. 


But  if  the  base  be  added  there  will  be  five  parts.  Therefore,  if 
the  complete  tonic  chord  is  required  it  will  be  necessary  to  omit 
some  note  from  the  discord.  If  we  leave  out  the  root  or  the  jth,  a 
dominant  yth  chord  will  no  longer  appear.  But  as  the  3d  and  the 
5th  each  resolve  to  the  tonic  we  may  omit  either  of  those  tones : 

*The  resolution  of  the  dominant  7th  chord  here  results  in  what  the  author  terms  a  half 
open  position. 

tThis  chord  is  also  known  as  the  Principal  7th,  and  as  the  Essential  7th. 


GOODRICH'S  ANALYTICAL  HARMONY. 


85 


Ex.  179. 


\<?        i  l 
JL_gjJ 


The  effect  upon  the  tonic  chord  is  the  same  whether  the  3d  or  the 
5th  be  omitted.  In  both  instances  the  concord  is  complete,  the  5th 
having  been  retained  from  the  duplicated  root-note  of  the  discord. 
The  combination  g,  d,  f  could  only  result  from  a  chord  founded  on 
G,  and  containing  b,  thus : 


Ex.  180. 


Likewise,  g,  b,  f  presuppose  d,  because  the  combination  could  not 
be  accounted  for  otherwise.  The  resolutions  of  the  discord  at  (a) 
and  (b)  of  Ex.  179  are  according  to  previous  directions,  which,  as 
they  are  important,  will  be  re-stated : 

The  root-note,  when  duplicated  above,  remains  stationary  as  5th 
of  the  tonic  chord ;  the  3d  ascends  to  the  tonic ;  the  5th  descends  to 
the  tonic;  the  yth  descends  to  the  3d  of  the  concord. 

The  3d  or  5th  of  the  discord  may  therefore  be  omitted,  and  with 
Ihe  best  results,  especially  when  the  tonic  chord  is  to  appear  in 
complete  form. 

How  may  the  discord  be  introduced  in  its  new  form  ?  This  is 
''nfluenced  by  the  contents  of  the  previous  chord.  If  the  3d  of  the 
liscord  appears  as  part  of  the  antecedent  chord,  it  is  better  to  retain 
1  hat  note  and  omit  the  5th  : 


Ex.  181. 


This  is  both  correct  and  effective.  Observe  the  chord  marked  + 
and  why  the  3d  is  included  to  the  exclusion  of  the  5th.  But  if  the 
antecedent  chord  contains  the  5th  of  the  discord,  it  will  be  better 
(for  the  same  reasons)  to  retain  that  note  and  omit  the  3d,  thus . 


86 


GOODRICH'S  ANALYTICAL  HARMONY. 


Ex.  182. 


The  5th  and  yth  of  the  discord  at  +  are  derived  from  the  previous 
chord,  and  do  not  progress.  By  duplicating  the  root  (in  order  to 
have  the  tonic  chord  complete)  the  3d  is  necessarily  omitted. 

Re-arrange  the  last  two  exercises  in  two  other  positions  and 
transpose  them  into  several  keys.  *  *  * 

There  is  another  circumstance  that  may  influence  the  omission 
of  a  certain  note  from  the  discord.  For  instance,  in  progressing 
from  the  sub-dominant  harmony  to  that  of  the  principal  yth,  the  5th 
is  omitted  to  avoid  parallel  5ths  : 

* 


Ex.  183. 


At  (a)  and  (b)  there  are  consecutive  5ths  between  the  base  and 
mezzo-soprano  parts.  The  5th  at  (c)  resolves  to  a  3d,  while  the 
upper  octave  remains,  and  becomes  yth  of  the  discord.  The  5th  of 
the  dominant  yth  chord  therefore  is  omitted.  Example  (c)  is  both 
correct  and  useful,  and  the  student  should  write  it  in  various  keys, 
with  the  other  arrangements  included,  thus : 


Ex.  184. 


Perform  all  these  illustrations  and  listen  to  their  effect.  If  the  dom- 
inant yth  chords  have  been  written  and  resolved  in  all  the  major 
keys,  as  advised,  it  will  not  be  necessary  to  make  extended  mechan- 
ical examples  of  the  discord  with  its  3d  or  5th  omitted,  as  the 
rules  of  resolution  remain  the  same.  If  the  task  has  been  neg- 
lected the  student  has  thereby  assumed  a  responsibility  which  v  ^i\d 
otherwise  have  rested  upon  the  author. 


JIL   "         ^s             ^r\ 

x     1   »      at'^-' 

';, 

PrS       £*"""« 

^_4_p==£l  — 

&         \           \ 

1 

—&  —  ^~ 

h*  *M 

\ 

GOODRICH  S    ANALYTICAL    HARMONY. 


A  theme  will  now  be  given  for  harmonization  in  which  the  sub- 
stance of  this  chapter  is  to  be  illustrated  : 
Ex.  185. 


s 


nt; 


£ 


As  the  base  is  fundamental  there  will  be  no  trouble  in  supplying 
the  harmony.  The  dashes  indicate  modulations,  and  the  chromatic 
notes  are  to  be  included.  |  means  that  the  5th  is  to  be  omitted  ; 
I  refers  to  the  omission  of  the  3d.  On  the  final  cadence  the  discord 
is  to  appear  in  complete  form,  with  its  3d,  5th,  and  yth,  and  resolved 
rulably.  When  completed,  this  theme  should  be  transposed  to  B- 
flat,  D  and  E-flat.  It  also  admits  of  re-arrangement,  and  this  should 
not  be  neglected,  for  it  shows  the  different  phases  of  a  certain  "pro- 
gression. A  solution  is  included  in  the  Key. 

As  a  summary  of  what  has  been  explained  in  this  chapter  the 
following  precepts  are  deduced  : 

1.  That  a  dominant  yth  chord  may  be  used  in  its  entirety,  or  it 
may  appear  with  its  3d  or  5th  omitted. 

2.  When  the  discord  appears  in  its  entirety,  and  is  resolved  cor- 
rectly, the  following  tonic  chord  will  appear  incomplete   (without 
its  5th). 

3.  When  the  3d  or  5th  of  the  discord  is  omitted,  and  the  root- 
note  is  duplicated  above,  it  resolves  to  the  tonic  triad  complete,  with 
the  5th  included. 

4.  That  the  5th  of  the  tonic  triad  is  to  be  included  in  all  inter- 
mediate passages. 

5.  That  in  the  final  cadence  the  full  dominant  yth  chord  may 
be  used  and  resolved  to  the  tonic  triad  without  its  5th. 

As  a  useful  practice  the  cadence  may  also  be  arranged  in  this 
manner,  by  altering  the  melody  : 


Ex.  186. 


.X)me  of  the  transpositions  might  end  in  this  way, 


88 


GOODRICH'S  ANALYTICAL  HARMONY. 


Compare  this  with  the  original,  Ex.  185 

Both  arrangements  are  correct  and  should  be  employed. 


An  important  but  rather  abstruse  principle  enters  here  with 
regard  to  tone  representations.  This  principle  will  be  adverted  to 
hereafter,  and  therefore  a  mere  elementary  phase  of  it  is  here  pre- 
sented. Under  ordinary  circumstances  the  final  tonic  chord  may 
appear  with  only  its  root  and  3d,  and  yet  the  effect  may  be  complete, 
as  though  the  5th  were  included : 


Ex.  187. 


m 


3 


The  explanation  lies  in  our  present  system  of  tonality.  These  two 
measures  embrace  every  note  in  the  C-major  scale,  and  as  the  theme 
naturally  leads  from  the  5th  up  to  the  tonic,  the  final  ending  upon 
C  is  anticipated.  Besides,  the  dominant  7th  chord  resolves  naturally 
to  the  chord  of  C-major. 

The  5th  of  this  chord  is  omitted  because  no  part  of  the  discord, 
as  here  arranged,  will  naturally  resolve  to  g.  But  the  mind  compre- 
hends this  note  as  part  of  the  scale. 

It  is  also  known  that  the  3d  or  5th  may  be  omitted  from  a  dom- 
inant 7th  chord  without  creating  an  incomplete  effect.  This  was 
explained  theoretically  upon  the  principle  of  fundamental  chord 
formation:  i,  5,  7  presuppose  3,  and  i,  3,  7  presuppose  5.  But  in 
addition  to  this  theorem  the  author  would  observe  that  precon- 
ceived ideas  of  tonality  have  become  so  fixed  that  we  readily  supply 
many  actual  omissions,  and  even  conceive  certain  harmonies  in  a 
relation  almost  totally  different  from  their  actual  preliminary  repre- 
sentation. 

An  illustration  that  frequently  occurs  in  composition  is  here 
cited  : 

J-      -N    .+? 

Ex.  188. 


GOODRICH 'S    ANALYTICAL    HARMONY.  8g 

At  (a)  the  tonic  chord  is  heard  with  its  5th  in  the  base,  and  as  this 
is  a  postlude  it  is  natural  to  expect  the  dominant  yth  founded  on 
the  real-base  to  follow,  as  it  does  at  (d).  But  at  (b)  the  notes  are  g, 
b,  e,  g,  and  these  constitute  the  triad  of  E-minor  according  to  theo- 
retical chord  formation.  But  never  for  a  moment  is  the  chord  at  (b) 
associated  with  the  triad  of  E-minor.  All  the  circumstances  tend 
towards  an  authentic  cadence  with  the  dominant  yth  as  a  basis,  and 
the  e  is  recognized  as  a  suspension  from  the  tonic  chord.  The  d  to 
which  the  e  resolves  is  also  anticipated,  even  without  the  7th  in  the 
base.  When  this  tone  is  heard  at  (c)  the  impression  is  confirmed. 
Under  such  circumstances  it  is  possible  to  omit  both  the  3d  and  5th 
from  an  essential  discord  without  leaving  the  mind  in  doubt  as  to 
the  harmonic  effect. 


Chapter  XXIII. 


MAJOR   AND   MINOR   RESOLUTIONS   OF  THE 
DOMINANT  SEVENTH   CHORD. 

ILLUSTRATIVE  THEMES. 

THE  fact  has  already  been  demonstrated  that   a   dominant  yth 
chord  is  founded  upon  the  dominant  of  a  major  or  a  minor 
scale.     Its  resolution  therefore  may  be  to  minor  as  well  as  to  major. 
As  the  resolution   to  tonic  major  is  more  natural,  if  not  more 
important,  it  is  numbered  i. 

The  resolution  to  tonic  minor  will  be  numbered  2.* 
Continue  to  follow  the  natural  order  as  to  major  or  minor  modes 
as  they  are  found  in  the  group  of  related   keys.     These  are  here 
presented : 

^  Q  *3 

:|_^      rg  ;fe=jg=pi^g==l 

zt=t=EE=±E3 


Ex.  189. 


&- 


*  Tonic  major  and  tonic  minor  refer  to  the  key  to  which  the  discord  naturally  belongs. 
Whether  the  tonic  be  major  or  minor  depends  upon  the  prevailing  tonality. 


GOODRICH'S  ANALYTICAL  HARMONY. 


The  connecting  notes  show  their  relation,  while  the  roots  in  the  base 
indicate  that  every  relative  minor  is  located  a  small  third  below  its 
major.  The  example  shows  the  related  keys  in  G,  and  at  the  same 
time  indicates  the  mode  of  each,  whether  major  or  minor. 

It  will  be  well  to  understand  this  tonal  principle,  even  beyond 
the  realm  of  these  nearly  related  keys.  A  corresponding  diagram  in 
E-minor  is  offered : 


Ex.  190. 


\~J/  9 


The  minor  keys  represented  by  minor  chords  appear  here  most 
prominently.  The  relative  major  of  each  is  shown  to  be  situated  a 
small  3d  above  its  minor.  All  these  keys  are  comprehended  in  the 
table  of  natural  or  primary  modulations.  They  are  so  intimately 
related  as  to  form  a  tonal  genu?,  or  union.  And  it  is  possible  to 
modulate  to  any  of  these  keys  and  return  to  the  original  without 
entirely  destroying  the  impression  of  the  main  tonic  or  initial  key. 

There  is  one  more  explanation  to  be  made  with  reference  to  the 
last  diagram.  The  chord  on  the  5th  of  the  scale  is  minor ;  but  this 
represents  B  as  tonic,  not  as  dominant.  So  long  as  we  remain  in  E- 
minor,  d-sharp  will  appear  in  place  of  d.  When  B  becomes  tonic  of 
a  related  key,  the  d-natural  will  be  the  natural  minor  3d. 

As  a  farther  illustration  of  the  minor  resolution  of  a  dominant 
yth  chord  a  short  exercise  is  given,  including  temporary  modulations 
to  each  of  the  related  minor  keys : 


Ex.  191. 


This  begins  and  ends  in  G-major,  passing  in  transit  through  the 
tonalities  of  the  three  related  minor  keys.  In  the  sixth  measure 
the  author  has  included  a  2  before  c,  because  c-sharp  is  embraced 
in  the  tonality  of  B-minor,  and  ckj  serves  to  restore  the  original  key- 
impression.  The  student  is  to  add  the  base  to  this,  using  only  the 
roots.  In  the  ninth  measure  the  dominant  of  the  original  key  is 
to  be  employed  as  fundamental  in  the  base. 

Re-arrange  this  in  two  other  positions,  and  transpose  into  F  and 
E-flat.     *    *    * 


GOODRICH'S  ANALYTICAL  HARMONY. 


Many  of  these  modulations  may  be  compassed  by  the  dominant 
chord  without  the  7th.  The  principal  exception  is  where  the  jih  is 
a  tone  foreign  to  the  prevailing  tonality.  Such  an  instance  is  the 
modulation  to  the  sub-dominant  from  any  major  key,  as  previously 
shown.  But  where  the  3d  of  the  dominant  chord  (leading- tone)  is  a 
chromatic  tone,  the  modulation  may  be  accomplished  without  the 
aid  of  the  7th.  Following  are  examples : 


Ex.  192. 


m 


Analyze  these  according  to  the  theory  here  outlined. 

The  7th  chords  have  been  introduced  partly  to  afford  the  student 
an  idea  of  their  theory  and  practice,  and  parti y  because  they  present 
variety  to  the  concords.  The  discord  is  especially  effective  where 
its  yth  is  derived  from  some  part  of  a  previous  concord.  Instances 
of  this  kind  have  already  been  introduced,  and  will  be  included  in 
future  exercises. 

The  next  melody  is  designed  principally  to  illustrate  the  first 
and  second  resolutions  of  the  dominant  7th  chord.  Considerable 
practice  is  required  in  arranging  and  transposing  such  examples,  for 
the  manner  in  which  we  arrive  at  a  discord  is  almost  as  important 
as  the  manner  of  departing  from  it : 


THEME. 


Ex.  193. 


The  modulations  are  indicated  as  usual,  and  where  the  7  is  in- 
cluded the  dominant  7th  is  to  be  used.  The  repeated  b-flat  in  the 
fourth  measure  indicates  a  modulation  back  to  the  principal  key, 
after  having  modulated  to  A-flat. 

The  temporary  transition  to  the  dominant  is  accomplished  with- 
out the  7th,  especially  as  this  progression : 

Ex-  '94-  E/L    g-^izj=j  js  better   than  this :    EX.  195. 

LA"  \J      i        I  I 


92  GOODRICH'S  ANALYTICAL  HARMONY. 

The  modulation  to  G-minor  includes  two  chromatic  tones.  At  least 
one  of  these  should  be  restored  in  the  last  ending.  This  may  be 
written  in  either  of  the  following  ways : 


Ex.  196. 


£ 


The  fact  must  be  remembered  that  the  modulation  to  G-minor,  as 
well  as  that  to  B-flat,  destroys  the  A-flat,  and  the  tonality  of  two 
flats  is  therefore  established  in  the  sixth  measure.  The  restoration 
of  this  element  of  transition  is,  accordingly,  a  necessity,  and  it  is  to 
be  considered  as  a  chromatic  tone. 

The  note  having  an  ?  under  it  may  be  accompanied  by  the  sub- 
dominant  harmony  or  by  the  dominant  yth. 

When  complete,  the  example  should  be  re-arranged  and  then 
transposed  into  D-flat  and  F.  The  student  must  become  familiar 
with  all  keys. 


Chapter  XXIV. 


FOUR  RESOLUTIONS  OF  THE  DOMINANT 
SEVENTH  CHORD  ANALYZED. 


TWO  other  resolutions  of  the  dominant  yth  chord  will  be  added 
to  those  already  explained.  The  first  resolution  is  to  tonic 
major,  the  second  is  to  tonic  minor.  (Tonic  here  refers  to  the  same 
key-tone ;  but  the  difference  in  signature  between  tonic  major  and 
tonic  minor  is  three  flats  or  three  sharps.  Relative  major  and  minor 
have  the  same  signature  but  different  key-tones.) 

The  third  resolution  is  to  the  minor  triad  situated  a  major  zd 
above  the  root  of  the  discord. 

The  fourth  resolution  is  to  the  major  triad  located  a  minor  id 
above  the  root  of  the  discord. 


GOODRICH'S    ANALYTICAL    HARMONY. 

They  are  here  represented  : 
Ex.  197. 


93 


~g"<srrr~g~2:g  —  r-g  —  a~r~g~£a:z] 


i  a  a  4 

The  first  two  are  familiar.  The  third  and  fourth  embrace  different 
principles.  The  discord  must  first  be  divided,  as  it  contains  two 
fifths.  The  lower  fifth  comes  first.  Both  notes  of  this  interval  can- 


not ascend  without  producing  parallel  fifths  :       Ex-  J98-  Hes= 


The  notes  must  therefore  resolve  in  opposite  directions,  as  oblique 
movement  is  not  here  possible  : 


Ex.  199. 


This  is  correct. 


The  upper  fifth  must  be  submitted  to  the  same  process,  with  this 

J    J 
result :     Ex.  200.  E: 


The  two  parts  may  now  be  combined :    Ex.  201. 


The  exception  to  this  resolution  occurs  when  the  yth  appears  in 
one  of  the  middle  parts.  This  will  be  illustrated  in  the  next  chapter. 

The  same  is  true  of  the  fourth  resolution.  Each  of  the  fifths 
nrist  resolve  contrarily.  Or,  we  may  resolve  the  lower  third  up  : 


Ex.  202. 


and  the  upper  third  down,  with  the  same  result : 


Ex.  203. 


The  student  may  now  proceed  to  write  on  one  staff  the  four 
resolutions  of  every  dominant  yth  chord.  (There  are  twelve,  be- 
sides the  enharmonic  equivalents.) 

Begin  each  example  on  the  dominant  of  a  maj6r  scale.  Number 
the  resolutions  i,  2,  3,  4,  and  do  not  forget  to  include  the  necessary 
chromatic  signs  for  2  and  4.  The  signature  of  the  first  resolution  is 
to  be  included  for  each  example,  thus : 


Ex.  204. 


m 


y.    \jj  Fg) — g- 


••  Observe  that  the  original  tonic  (a  4th  above  the  root  of  the  discord)  appears  in  all  of  the 
resolutions. 


94  GOODRICH'S  ANALYTICAL  HARMONY. 

Begin  the  next  example  with  the  dominant  yth  on  A,  key  of  D ; 
then  three  sharps,  and  so  on  till  the  cycle  is  completed.* 

Five,  six  and  seven  sharps  ought  to  be  written  enharmonically. 
One  of  these  is  included  as  a  guide : 

a          ,      I  3     ,      .      I  *     , 

^  _   ^ 

**  -&--&--»--&• 

Ex.  205. 


THE  RESOLUTIONS  ANALYZED. 

Numbers  i  and  2  we  will  call  Principal  or  Regular  Resolutions. 
In  both  instances  the  elements  of  transition  are  resolved  rulably. 

Numbers  3  and  4  involve  a  new  principle  in  our  theory.  The 
elements  of  transition  consist  of  the  leading-tone  and  sub-dominant. 
These  elements  must  occur  as  parts  of  the  resultant  concord.  Ex- 

IL 

amine  the  third  resolution  :   Ex-  2o6<  F/K  *~&-~*%^:L = 

The  sub-dominant  to  e  is  a.  This  appears  in  the  discord,  and  its 
resolution  to  g  is  perfectly  regular.  But  the  leading-tone  to  e  is 
d-sharp,  and  the  root  of  the  discord  is  d-natural.  This  sub-tonic 
could  not  occur  in  a  direct  transition  nor  in  an  authentic  cadence. 
We  therefore  name  this  a  Secondary  or  Irregular  Resolution. 

In  the  fourth  resolution  the  conditions  are  reversed  in  respect  of 
the  elements  of  transition.  The  leading-note  is  present,  but  in  place 
of  the  sub-dominant  (a-flaf)  we  have  a-natural.  The  key  can  not 
be  decided  as. that  of  E-flat,  for  there  are  two  notes  foreign  to  that 

it 

key:    Ex.  207. {zj^,jz:g:zj^zz        Consequently  this,  like  No.  3,  is  an 

ogzr—g-fr-ggizJ 

Irregular  Resolution.  In  other  words,  Nos.  3  and  4  are  to  be  con- 
sidered rather  as  progressions,  and  not  as  determining  the  keys 
represented  by  the  resulting  concords. 

This  will  prove  true  of  every  example,  i  and  2  are  direct  reso- 
lutions, and  may  terminate  a  period  or  a  composition.  3  and  4  are 

*These  theoretical  examples  must  be  kept  together,  as  they  will  be  required  in  the  r<xt 
chapter. 


GOODRICH  S    ANALYTICAL    HARMONY.  05 

incomplete,  indirect  resolutions,  and  something  more  is  expected  to 
follow. 

The  student  should  analyze  several  examples  as  indicated.  The 
preliminary  task  is  a  mere  theoretical  exercise.  The  applications 
and  classifications  belong  to  the  next  chapter. 


Chapter  XXV. 


FOUR  RESOLUTIONS  OF  THE  DOMINANT 

SEVENTH    CHORD   CLASSIFIED 

AND   CHARACTERIZED. 

DIRECT  AND  AVOIDED  CADENCES. 

E  four  resolutions  of  a  principal  discord  will  now  be  classified 
-*•    with  a  view  to  their  application  in  actual  composition. 

In  the  key  of  F-major  how  many  of  the  resolutions  of  the  domi- 
nant yth  chord  can  be  used  naturally?  In  order  to  answer  the 
question  it  will  be  necessary  to  imagine  the  concords  in  this  key, 
namely,  F,  G,  A,  B-flat,  C  and  D.  The  discord  may  therefore  re- 
solve naturally  into  either  of  these  two  chords : 


Ex.  208. 


No.  2  is  to  F-minor ;  No.  4  is  to  D-flat-major.  Both  these  chords 
are  unnatural  in  the  key  of  F-major.  But  the  discord,  c,  e,  g,  b  -flat, 
belongs  to  another  key,  namely,  F-minor.  Write  the  second  and 
fourth  resolutions  in  this  key,  and  compare  the  result : 

9E    •'  * 


Ex.  209. 


By  referring  to  the  consonant  chords  in  F-minor  we  may  ascertain 
if  the  last  two  resolutions  are  practicable : 

-Or-**- 


Ex.  210. 


96  GOODRICH  S    ANALYTICAL    HARMONY. 

The  two  concords  (Ex.  209)  occur  here  naturally.  From  this  wet 
may  conclude  that  in  the  key  of  J:-/uhio>-  the  second  or  fourth  reso- 
lutions of  the  dominant  7th  chord  might  be  used,  but  that  in  the 
Key  of  F-major  these  resolutions  would  be  unnatural,  and  i  or  3 
snould  be  selected.  In  either  mode  a  direct  or  an  avoided  cadence 
may  be  chosen.  In  the  two  examples  thus  far  classified  it  should 
be  noted  that  the  resulting  triads  appear  naturally,  without  chro- 
matic signs. 

The  student  should  now  classify  all  the  examples  of  the  previous 
lesson.  Write  the  first  and  third  resolutions  in  the  key  of  the  tonic- 
major  first : 

3 


These  belong  to  C-major.     Then  write  the  second  and  fourth  reso- 
lutions in  the  key,  and  with  the  signature  of  the  tonic-minor : 


EX.  212. 


These  belong  naturally  to  C-minor. 

The  four  resolutions  of  the  dominant  yth  chord  on  G  appear  in 
the  two  examples,  but  as  the  discord  belongs  equally  to  two  differ- 
ent modes,  the  first  and  third  resolutions  occur  naturally  in  C-major; 
whereas  the  second  and  fourth  are  associated  with  the  tonality  of  C- 
minor. 

As  C-major  is  a  natural  scale,  and  as  C-minor  has  three  flats,  the 
signatures  are  too  much  at  variance  to  admit  of  intimate  relationship. 

The  student  is  expected  to  complete  the  classifications  in  the 
manner  illustrated.  *  *  * 

Having  ascertained  the  peculiar  tonal  characteristics  of  the 
various  concords  into  which  a  principal  discord  many  disappear,  the 
next  step  is  to  know  the  objects  of  these  different  kinds  of  resolu- 
tion. 

The  first  and  second  constitute  Direct  (Authentic)  Cadences ; 
the  third  and  fourth  constitute  Avoided  Cadences. 

In  a  direct  cadence  the  discord  is  resolved  in  a  natural  and  de- 
emed manner. 

In  an  avoided  cadence  the  discord  is  resolved  in  an  unnatural, 
undecided  manner. 


GOODRICH'S  ANALYTICAL  HARMONY.  97 

The  object  of  a  direct  cadence  is  to  decide  a  key  or  close  a 
period. 

The  object  of  an  avoided  cadence  is  to  prevent  a  final  close,  and 
thus  prolong  the  period.  One  is  determinate,  the  other  is  indeter- 
minate. 

The  direct  resolution  is  most  useful  at  the  close  of  a  period  or 
movement.  The  indirect  resolution  serves  the  best  purpose  in  the 
middle  of  a  period,  or  before  the  last  ending,  as  a  means  of  sustain- 
ing the  interest  by  leaving  the  ear  unsatisfied  until  the  direct  ca- 
dence occurs. 

The  number  of  cadences  to  be  employed  in  any  tonal  genus 
must  now  be  ascertained.  The  previous  classification  will  help  to 
solve  this  problem.  The  tonal  genus  comprehends  the  tonality  of 
the  six  keys  represented  by  these  familiar  concords  : 

Ex.  213. 


The  resolutions  of  the  discord,  whether  direct  or  indirect,  must,  for 
the  present,  be  into  some  of  these  triads.  The  dominant  yth  chord 
that  represents  the  first  triad  is  founded  on  the  top  note  of  the  triad. 
Write  this.  *  *  * 

Of  the  four  resolutions  of  this  discord  how  many  can  be  used 
naturally  in  this  key  ?  Mention  the  chords  and  give  their  numbers 
in  relation  to  the  four  resolutions?  Write  these.  Which  is  an 
avoided  and  which  a  direct  cadence  ?  *  *  *  (This  has  already 
been  illustrated  in  Exs.  211  and  212.)  Write  the  dominant  yth  chord 
that  represents  the  second  triad.  How  many  resolutions  of  this  dis- 
cord can  be  employed  naturally  in  this  key?  By  referring  to  the 
original  example  in  D,  it  will  be  seen  that  the  four  resolutions  of 
\vhich  the  discord  is  capable  are:  i,  D-major ;  2,  D-minor ;  3,  B- 
minor ;  4,  B-flat-major.  Only  one  of  these  can  be  used  in  the 


present  instance:      x  2I4  \^p — £MZ        This  is  a  direct  resolution. 


Write  the  dominant  yth  chord  that  represents  the  third  triad. 
The  fifth  of  the  E-minor  triad  is  b  :  a  principal  discord,  founded  on 
B,  will  contain  d-sharp,f-sharp,  and  a,  besides  the  root.  An  exam- 
ination of  the  original  example  in  E,  which  is  here  used  as  a  chart, 
will  reveal  the  fact  that  the  four  resolutions  of  the  discord  on  B  are : 


98  GOODRICH'S  ANALYTICAL  HARMONY. 

i,  E-major ;  2,  E-minor ;  3,  C-sharp-mi nor  ;  4,  C-major.  J.t  is  evident 
that  2  and  4  answer  present  purposes.  "Write  these,  and  keep  all 
examples  together. 

The  discord  representing  the  fourth  triad  comes  next.  After 
writing  this,  inquire  what  are  its  four  resolutions,  and,which  will  be 
proper  to  use  in  this  key.  Here,  likewise,  is  a  direct  and  an  avoided 
cadence. 

The  discord  on  D,  representing  the  fifth  triad,  is  next  in  order. 
Carry  out  the  same  formula  as  to  questions  and  answers. 

Next  select  the  dominant  yth  chord  on  E,  representing  the  sixth 
triad,  A-minor,  and  the  examples  will  be  completed.  This  has  a 
direct  and  avoided  cadence  in  this  key.  *  *  * 

The  available  resolutions  of  a  dominant  yth  chord  have  been 
employed  on  every  tone  of  the  scale  excepting  the  fourth.  This 
does  not  represent  a  related  key  in  any  of  its  resolutions,  but  belongs 
either  to  B-flat  major  or  B-flat  minor.  Hence  its  rejection  here. 

The  student  is  to  remember  that  though  the  resulting  concords 
occur  naturally  and  are  not  altered,  the  discords  require  chromatic 
alteration  because  they  are  transition  chords  in  their  original  ap- 
plication. This  is  true  of  all  except  the  discord  to  the  central,  or 
principal  key,  which  belongs  to  its  own  scale  and  occurs  naturally. 

Complete  examples  are  to  be  written  in  D,  E,  B-flat,  A-flat  and 
G-flat.  These  will  be  required  in  the  next  chapter. 


Chapter  XXVI. 


AVOIDED  CADENCES  ILLUSTRATED. 


>~rVHE  avoided  cadences  previously  classified  and  characterized  are 


to  be  applied  as  in  actual  composition : 

CHART. 

3  „   "i 


GOODRICH'S  ANALYTICAL  HARMONY. 


99 


By  omitting  the  direct  cadences  d  and  2)  it  will  be  a  simple  task 
to  extract  from  this  chart  the  avoided  cadences  (3  and  4),  and  this 
should  now  be  done.  *  *  * 

The  next  step  is  to  ascertain  how  these  are  to  be  arranged  with 
regard  to  their  harmonic  progression.  The  first  position  of  the  dis- 
cord is  not  favorable  to  an  indirect  resolution,  for  in  order  to  avoid 
false  fifths  it  is  necessary  to  double  the  3d  of  the  concord,  and  this 
leaves  the  latter  in  an  abbreviated  form : 


Ex.  216.    <' 


This  is  not  here  recommended.     Invert  the  upper  parts,  as  the  base 
can  not  be  altered: 


Ex.    217. 


The  3d  of  the  discord  descends,  though  in  the  original  position  it 
was  compelled  to  ascend,  to  prevent  parallel  fifths  with  the  f  above. 
But  here  the  5th  (b  and  /)  appears  as  a  4th  (/and  £),  and  there  is 
no  prohibition  against  consecutive  fourths  when  they  are  accom- 
panied by  another  interval.  This  example  is,  therefore,  correct.  The 
7th  must  descend  to  the  5th  of  the  concord ;  the  5th  of  the  discord 
is  the  sub-dominant,  and  must  descend  to  the  3d  of  the  triad  as 


though  it  were  the  yth  of  the  dominant : 


Ex.  218. 


The  root  of  the  discord  in  the  base  must  ascend  a  2d  to  the  root 
of  the  triad. 

These  are  the  most  important  resolutions,  and  from  these  direc- 
tions there  will  be  no  deviation.  The  resolution  of  the  3d  of  the 
discord  is  variable.  When  it  is  below  the  7th  it  must  ascend,  when 
above  the  yth  it  may  descend.  This  latter  plan  will  be  adopted  here.* 

*  As  a  discord  may  disappear  in  many  ways,  the  author  does  not  intend  to  issue  instruc- 
tions except  for  particular  instances.  These  rules  merely  apply  to  the  third  and  fourth  reso- 
lutions of  an  essential  discord. 


joo  GOODRICH'S  ANALYTICAL  HARMONY. 

Another  position  of  the  discord  is  selected  for  resolution : 


Ex.    219. 


Each  note  of  the  discord  is  here  resolved  the  same  as  in  Ex.  217, 
which  see.  No  false  progressions  result ;  the  upper  parts  move  in 
an  opposite  direction  to  that  of  the  base,  and  the  example  is  correct. 
At  present  only  these  two  arrangements  will  be  used,  i.  e.,  with  the 
3d  or  5th  uppermost.  These  positions  and  their  accompanying 
directions  will  apply  to  every  discord  of  the  dominant  7th  when 
resolved  indirectly. 

Herewith  a  few  indirect  resolutions  are  presented,  as  they  are  to 
be  applied  in  the  harmonizations  that  follow : 


EX.  220. 


3. 


3 


The  first  two  measures  are  third  resolutions;  the  next  two  are 
fourth  resolutions.  All  are  good.  In  the  fourth  resolution  one  of 
the  upper  parts  moves  down  an  augmented  2d.  This  is  correct, 
according  to  the  testimony  furnished  by  the  most  eminent  compos- 
ers. Here  are  two  corroborative  illustrations  from  the  immortal 
Beethoven : 


Op.  10,  No.  2.  b. 


-  Op.  53. 
"2- 


EX.  221. 


At  (a)  the  augmented  2d  appears  ascending  and  descending,  and  at 
(b)  the  b-sharp  descends  plainly  enough  to  a. 


*The  last  resolution  will  be  utilized  iu  Harmonic  Counterpoint. 


GOODRICH'S  ANALYTICAL  HARMONY. 


101 


Ex.   222. 


THEME    FOR    HARMONIZATION. 
«•    .  _***»  *„—  _ 


.0..f    0 


~f -j£ 

-F — F — &- 

L-LX 


The  dashes  show  where  avoided  cadences  take  place.  The  chart 
containing  the  five  avoided  cadences  in  this  tonal  genus,  together 
with  such  aid  as  the  theme  affords,  will  enable  the  student  to  deter- 
mine upon  the  proper  chords.  All  five  irregular  resolutions  are  to 
be  employed,  and  that  one  which  avoids  the  tonic  cadence  to  C  is  to 
be  used  twice.  The  last  cadence  marked  +  is  to  be  direct  and  final. 

The  author  will  repeat  the  directions  here,  in  order  to  free  them 
from  previous  explanations  and  examples  : 

1.  The  indirect  resolution  must  be  to  some  of  the  related  con- 

cords. 

2.  The  melody  note  is  to  be  considered  as  3d  or  5th  of  the 

essential  discord  whenever  an  avoided  cadence  is  made. 

3.  The  base  is  to  ascend  a  2d  from  root  to  root. 

4.  The  upper  parts  must  descend  in  order  to  form  contrary 

movement  to  the  base. 

5.  The  root-note  of  the  discord  is  not  to  be  duplicated  in  any  of 

the  upper  parts. 

6.  The  jth  of  the  discord  must  not  (at  present)  appear  in  the 

melody. 

Students  are  not  required  to  commit  these  directions  to  memory, 
but  to  understand  the  principles  involved.  To  do  so  it  may  be 
necessary  to  refer  back  to  the  illustrative  examples,  for  every  direc- 
tion has  been  duly  exemplified.  The  time  has  come  for  us  to  dis- 
card the  machine  methods  of  the  school-room,  where  teachers  are 
still  groping  in  the  dark,  oblivious  of  the  fact  that  cramming  the 
memory  does  not  cultivate  the  mind,  or  that  one  may  "  memorize 
lessons  "  without  comprehending  them. 

•  A  few  of  the  progressions  in  the  last  example  may  cause  mis- 
givings on  the  part  of  the  student,  as  where  no  connection  appears 
in  the  upper  parts,  and  where  the  base  moves  more  than  one  degree : 


Ex.  223. 


c 


_    _ 


102 


GOODRICH'S  ANALYTICAL  HARMONY. 


While  the  progression  at  (a)  is  not  positively  wrong,  it  has  some- 
what the  appearance  of  evil  on  account  of  the  similar  movement  of 
all  the  parts.  The  arrangement  at  (b)  is  preferable,  and  should  be 
employed  when  it  is  possible. 

The  harmonization  of  the  theme  should  now  be  completed. 

*     *    * 

Almost  every  good  composition  verifies  this  application  of  the 
four  resolutions  of  a  dominant  yth  chord.  In  fact,  this  system  is 
based  upon  actual  composition,  not  upon  mathematical  theories  or 
vague  hypotheses. 

The  harmonization  of  the  last  theme  will  be  presented,  as  it  con- 
tains some  features  comparatively  new  to  the  student : 
Ex.  224. 


The  E-minor  triad  is  perhaps  better  in  the  first  and  fifth  meas- 
ures, as  it  furnishes  a  connecting  note  with  the  following  discord. 
Observe  the  contrary  movement  in  such  places  as  from  the  third  to 
the  fourth  chords,  first  measure.  The  last  avoided  cadence  might 
have  been  arranged  without  the  yth,  in  thi?  way, 


Ex.  225. 


etc. 


especially  as  the  avoided  cadence  to  A-miw  WPS  included  in  the 
first  of  the  example. 

The  last  complete  cadence  might  also  have  been  written  Jik? 
this: 


Ex.  226. 


2      <£: 

m            ^\ 

—  0  —  &•  — 

1 

—  1  1.  ..._. 

~*~\ 

In  such  instances  the  yth  is  not  absolutely  essential,  though  the 


GOODRICH'S  ANALYTICAL  HARMONY. 


connecting  note  gives  more  consistency  to  the  harmony,  and  is  gen- 
erally preferable. 

The  theme  should  be  transposed  to  B-flat,  D,  E-flat,  and  F, 
Then  write  the  avoided  cadences  associated  with  each  key  and  pro- 
ceed with  the  harmonizations. 

In  one  of  these  examples  the  tonic  triad  may  be  substituted  for 
the  minor  triad  on  the  mediant.  The  last  example  does  not  admit 
re-arrangement. 

The  third  resolution  of  a  dominant  yth  chord  is  inclined  to  be 
plaintive,  to  express  disappointment  and  regiet.  Schubert  has  em- 
ployed it  in  this  sense. 

The  fourth  resolution  is  bolder  and  brighter,  though  generally 
unexpected. 

Both  cadences  have  this  in  common  :  They  avoid  the  regular 
cadence,  and  thus  serve  to  postpone  the  ending ;  or  preserve  the  in- 
terest, instead  of  allowing  it  to  subside. 

For  most  interesting  illustrations  of  avoided  cadences,  the  reader 
is  referred  to  the  Romance  from  Tannkduser, "  O,  evening  star." 
The  first  five  cadences  are  avoided  by  means  of  third  and  fourth 
resolutions  of  the  dominant  7th  chord. 


Note.  Numerous  instances  occur  in  which  a  fourth  resolution  is  used 
differently  than  in  the  author's  classification.  These  occur  in  the  nature  of 
abrupt  transitions,  and  usually  after  the  prevailing  tonality  has  been  more  or 
less  exhausted.  Under  these  circumstances  the  ear  more  readily  follows  any 
unusual  progression. 

Rossini  was  very  partial  to  this  expediency — so  much  so  that  he  employed ' 
it  in  nearly  all  his  operatic  finales  and  Overtures !  See  Overtures  to  Othello 
(after  the  return  of  the  first  subject); — Tancredi  (the  finale).  Semaramis 
(end  of  first  subject). 

An  illustration  from  a  popular  overture  is  quoted.  The  fourth  resolution 
occurs  after  several  periods  in  A-major  : 

Herald. 
Ex.  227. 


The  strain  beginning  in  F  is  the  same  as  that  of  the  preceding  in  A,  but 
after  the  avoided  cadence  it  appears  in  a  new  light. 


-04 


GOODRICH  S   ANALYTICAL    HARMONY. 


PART  VI. 


Chapter  XXVII. 


INVERTED  BASES. 

THEIR  OBJECT  AND  EFFECT. 

THE  various  close  positions  of  major  and  minor  concords  and 
the  re-arrangements  of  the  dominant  yth  chord  have  been  pre- 
sented and  explained.  It  has  also  been  shown  that  a  concord  or  a 
discord  has  as  many  positions  as  notes. 

Heretofore  the  movement  of  the  base  has  invariably  been  fun- 
damental, from  root  to  root.  In  modern  music,  however,  the  base 
is  considered  equally  important  with  the  other  parts,  and  it  may 
therefore  assume  any  position  in  a  chord  that  its  melodic  progression 
requires. 

The  word  Inversion  has  frequently  been  applied  to  intervals  in 
the  previous  lessons,  but  in  an  Inverted  chord  the  base  has  some 
other  tone  than  the  root.  "  Real-base, "  or  actual  base,  serves  to 
designate  the  lowest  part  of  the  harmony,  and  also  to  indicate  that 
the  base  note  is  not  a  root-note.  The  simplest  example  occurs  when 
the  base  part  executes  different  intervals  of  a  chord,  thus : 


Mozart. 


Ex.  228. 


The  solo  in  the  base  part  consists  of  a  chord  motive.     The  othei 
parts  (violins,  etc.)  merely  accompany  the  solo  with  the  harmony  of 


GOODRICH'S  ANALYTICAL  HARMONY. 


105 


D,  indicated  by  the  chord  figure  below.  This  idea  is  continued 
during  the  first  sixteen  measures  of  the  Symphony  (No.  23,  B.  and 
H.)  and  it  would  be  well  for  students  to  examine  and  perform  the 
entire  passage. 

A  similar  instance  occurs  at  the  end  of  periods  in  popular  music, 
where  the  base  passes  through  the  different  tones  of  the  tonic  chord 
while  the  upper  parts  sustain  the  same  harmony : 


Jos.  Strauss. 


Ex.  229. 


^ 


•=tr. 


X    X 


(2)    (1) 

The  figures  (2)  (i)  indicate  the  inversions.  This  is  so  evident  that 
no  farther  explanations  seem  necessary. 

The  management  of  inverted  bases  requires  considerable  practical 
experience  and  theoretical  information,  and  for  this  reason  their  in- 
troduction has  for  so  long  a  time  been  deferred. 

When  the  base  has  the  3d  of  the  triad  it  is  customary  to  omit 
that  note  from  the  upper  parts,  thus : 


Mozart. 


Ex.  230. 


The  soprano  skips  up  a  3d,  thus  forming  a  counter  melody  to  the 
base.  The  other  parts  remain  stationary,  as  may  be  seen  by 
arranging  the  example  in  this  manner: 


Ex.  231. 


(D  (D 

Aside  from  the  two  melodic  parts,  which  here  result  from  changing 
positions,  the  principal  reason  for  omitting  the  3d  above  when  it 


io6 


GOODRICH'S  ANALYTICAL  HARMONY. 


appears  below  is,  that  this  interval  determines  the  character  of  the 
chord  (whether  major  or  minor),  and  on  account  of  its  strength  it 
becomes  too  prominent  if  doubled  above. 


If  this  interval  Ex-  *32-  P/U   J  —  appears  to  create  a  void  in  the 


harmony  on  account  of  its  ambiguity,  the  ear  will  experience  an 
agreeable  sensation  in  discovering  the  characteristic  tone  below, 
which  completes  the  tonal  effect : 


Ex.  233. 


Perform  the  examples  separately  and  listen  to  the  effect.  This 
characteristic  quality  .of  the  3d  is  stronger  in  major  than  in  minor 
chords. 

Another  reason  for  omitting  the  3d  above  when  it  occurs  in  the 
base  is,  that  if  the  3d  be  doubled  this  duplication  is  liable  to  result 
in  false  progressions,  on  account  of  the  tendency  of  the  two  thirds 
to  move  in  parallel  movement.  This  is  especially  true  in  chord 
progression  where  the  parts  move  alphabetically : 


Ex.  234. 


This  results  in  consecutive  octaves  between  the  extreme  parts,  and 
should  be  avoided.  There  are  several  ways  in  which  the  same 
chords  and  the  same  base  may  be  arranged  correctly : 


Ex.  235. 


4j-«  C           J— 

^K      ^F 

=s  — 

i 

*/       s* 

0 

h""\*  T 

i 

i 

~J±t!  — 

=q 

^      J 

— 

— 

-*  —  I 

(i) 

(i) 

u 

I 

I  —  \- 

rr  ,   i 

\        , 

—  ff— 

4 

—  1(- 

-*- 

—  H  •-- 

2 

j 

5        ^ 

• 

*        * 

*    *  * 

1 

, 

-  1 

! 

j 

J             1 

,~*~  *~ 

-/- 

—  4- 

*      '      1 

(i) 


GOODRICH  S   ANALYTICAL    HARMONY. 


IO7 


The  inversion  is  accompanied  by  a  half-open  position  of  the  G 
chord,  as  a  convenient  method  for  avoiding  the  duplicated  3d.  In 
the  second  measure  the  soprano  and  contralto  parts  move  together 
at  the  distance  of  an  octave,  producing  a  counter-theme  to  the  base. 
These  octaves  are  not  objectionable ;  in  fact,  both  arrangements  are 
decidedly  preferable  to  Ex.  234.  The  only  caution  necessary  is  this : 
if  these  half-open  positions  be  continued  beyond  the  influence  of  the 
connecting  tone  (g,  in  last  example)  they  will  generally  result  in 
evil. 

The  fifth  of  a  concord  as  real-base  now  claims  attention.  Nothing 
more  need  be  said  of  bases  that  merely  pass  through  the  different 
tones  of  an  unchanging  harmony,  except  that  such  instances  are 
numerous  and  effective. 

In  Progression,  when  the  base  occupies  the  fifth  of  a  concord, 
the  conditions  are  altered,  and  some  care  is  required  in  its  manage- 
ment. 

There  is  an  old  thorough-base  law  to  this  effect,  that "  a  f  chord 
must  be  followed  by  the  dominant  or  dominant  7th  harmony."  This 
signifies  that  when  the  5th  of  a  concord  is  in  the  base,  the  latter 
remains  and  becomes  the  root. 


Ex.  236. 


The  5th  of  the  C  chord  in  the  base  is  somewhat  out  of  balance,  and 
the  following  dominant  chord  serves  to  restore  the  equilibrium ;  the 
base  remaining  as  root  and  connecting-tone.  This  forms  a  part  of . 
the  perfect  cadence,  as  will  be  seen  later. 

The  formula  just  quoted  has  been  much  used  by  composers,  and 
though  not  now  followed  so  literally  as  it  once  was,  the  student  can 
not  do  better  than  adopt  -it  until  some  other  method  is  offered.  The 
same  directions  apply  to  both  modes. 

The  inversion  of  the  chord  of  the  dominant  7th  is  next  in  order. 
Any  of  its  tones  may  occur  in  the  base.  Therefore  the  base  is  said 
to  be  inverted  when  the  3d,  5th,  or  7th  of  the  discord  appears  below 
instead  of  above.  In  each  of  these  instances  the  base  note  is  to  be 

*This  might  have  been  the  dominant  ?th. 


io8 


GOODRICH'S  ANALYTICAL  HARMONY. 


omitted  from  the  upper  parts.     The  fundamental  position  may  ap- 
pear ia  any  of  these  forms  : 


Ex.  237. 


The  first  inversion  is  to  be  written  with  the  3d  omitted  above ;  so 
with  the  second  inversion,  and  especially  with  the  third,  in  which 
the  7th  is  below : 


Ex.  238. 


^EE 


(1)        (2)       (3) 

Observe  that  in  each  position  the  discord  appears  complete.  The 
notes  omitted  above  are  supplied  by  the  real-base  below.  Each  of 
these  measures  is  capable  of  being  re-arranged  in  three  positions : 


Ex.  239. 


_^  +  +  —  *__ 

-*  •*-*!•  —  • 

«    i* 

5 

J        J       1 

€      J      • 

-9-      '              -9-      '       1 

^2     * 

—  t  —  (&  —  —  

i  

1 

(1) 


(2) 


(3) 


Compare  each  upper  position  with  the  base. 

Throughout  all  these  inversions  and  re-arrangements  the  root,  of 
fundamental  remains  C,  the  theoretical  generator  of  the  discord. 

The  resolutions  of  these  inversions  must  now  be  undertaken. 
The  directions  remain  in  force  so  long  as  the  resolutions  are  to  tonic 
major  or  minor. 

The  3d  must  be  resolved  up  a  2d,  and  the  yth  down  a  2d  without 
regard  to  the  position  of  the  discord.  The  5th  usually  resolves  down 
to  the  tonic.  The  only  difference  between  base  and  treble  parts  is 
this,  that  the  root  in  the  base  ascends  a  4th  or  descends  a  5th,  where- 
as the  duplicated  root-note  in  any  of  the  upper  parts  remains  station- 


The  figures  i,  2,  3  refer  to  the  number  of  the  inversion. 


GOODRICH  S    ANALYTICAL    HARMONY. 


109 


ary  as  a  connecting  link.     The  student  many  now  proceed  with  the 
resolutions,  of  which  a  sample  is  given  : 
Ex.  240. 


Minor. 


(1)  (2)  (3)  (I)  (I)  (2)  (3)  (1)     (3)  (!) 

With  exception  of  the  inverted  bases  there  is  nothing  new  in  these 
resolutions.  The  third  measure  of  the  minor  example  shows  a  half- 
open  position  of  the  concord.  But  this  merely  results  from  the 
regular  resolution  of  the  3d  and  the  5th  (e-natural  and  g}  to  the 
tonic,  as  in  the  third  measure  of  the  major  example.  The  contrary 
movement  in  the  last  measure  is  also  good,  provided  the  following 
progressions  are  in  keeping  with  it. 

Several  examples  similar  to  the  last  ought  to  be  completed, 
some  of  which  should  contain  resolutions  of  the  re-arrangements. 
See  Ex.  240.  The  third  inversion  is  the  most  troublesome  to  man- 
age proper!}-,  because  its  resolution  leaves  the  concord  either  in  an 
abbreviated  form  as  here, 


Ex.  241. 


or  in  a  half-open 
position  like  this : 


Ex.  242. 


m 


(3)     (1) 


(3)     (I) 


In  either  case  the  inexperienced  harmonist  would  be  liable  to  en- 
counter some  difficulty  in  progressing  beyond  the  second  chord, 
which  is  likewise  inverted.  For  the  purpose  of  anticipating  these 
difficulties  a  number  of  examples  are  offered  in  which  the  yth  ap- 
pears as  real-base : 
Ex.  243. 


no 


GOODRICH  S    ANALYTICAL    HARMONY. 


The  first  two  exercises  embrace  a  modulation  to  the  sub-dominant 
and  back  to  the  original  tonic.  They  are  alike,  excepting  the  treat- 
ment of  the  C  chord  with  its  5th  as  real-base.  This  scheme,  with 
various  modifications,  has  been  much  used.  At  (b)  there  is  no  ob- 
jection to  the  duplicated  5th  in  the  C  chord.  Example  (c)  embraces 
a  transition  to  the  dominant.  This  contains  two  peculiarities.  The 
C  chord  is  followed  by  the  D  chord,  each  in  the  same  position.  Hut 
as  no  fifths  appear,  and  as  the  base  acts  as  connecting  note,  no  objec- 
tion can  be  raised  against  this.  The  second,  c  in  the  base  becomes 
7th  of  the  chord  on  D,  and  this  is  resolved  correctly,  although 
different  in  one  respect  from  the  other  resolutions  of  a  thiid  inver- 
sion. The  5th  of  the  discord  ascends  to  the  3d  of  the  concord,  thus 
doubling  that  tone  which  the  base  is  obliged  to  sound.  But  a  good 
reason  appears  for  the  duplicated  3d.  The  mezzo-soprano  part  has 
a  regular  melodic  progression,  g,  a,  b,  c,  which  accords  well  with  the 

base.     These  two  parts  are  presented : 

,       i 

J 


1     I 

In  separating  from  each  other  they  sound  b  simultaneously,  and  as 
this  was  necessary  to  the  design,  the  temporary  prominence  of  the 
major  3d  in  the  middle  of  the  progression  is  not  objectionable.  Be- 
sides, the  other  parts  contribute  to  the  good  effect. 

An  important  esthetic  principle  is  here  enunciated :  the  regular 
melodic  progression  of  any  voice-part  may  justify  the  most  severe 
dissonances  or  the  most  unusual  harmonic  progressions  which  would 
otherwise  be  intolerable.  Sequences  also  justify  many  transgres- 
sions of  grammatical  rules  and  harmonic  precepts.  The  following 
extract  may  be  explained  in  the  same  manner : 


A.  Scarlatti. 


Ex.  245. 


The  base  has  a  regular  melodic  progression  upward,  so  have  the 
other  parts  downward.  That  a  duplicated  3d  and  duplicated  5th 
result,  is  not  to  be  objected  to,  for  the  design  is  more  important  than 
the  preservation  of  an  arbitrary  formula. 


OOODRICH'S  ANALYTICAL  HARMONY. 


in 


Another  method  of  employing  inverted  bases,  and  one  that  is 
more  easily  reduced  to  practical  theory,  is  the  following:  Whenever 
a  modulation  is  made  to  the  key  a  third  below,  the  tone  between 
the  two  tonics  may  be  given  the  base.  This  tone  is  to  be  the  5th 
of  the  transition  chord,  and  must  of  course  be  omitted  from  the 
upper  harmony. 

Suppose  the  pupil  is  writing  in  F-major  and  wishes  to  modulate 
to  the  3d  below.  In  elementary  modulation  the  base  moved  from 
root  to  root : 


Ex.  246. 


m. 


But  the  tone  between  F  and  D  being  a  part  of  the  dominant  yth 
chord  to  D,  may  appear  in  the  base,  thus : 


Ex.  247. 


This  gives  to  the  base  a  distinct  melodic  progression,  and  is  a  con- 
siderable improvement  upon  Ex.  246.  With  all  these  transitions  to 
the  3d  below,  this  second  inversion  may  be  used,  and  in  every  in- 
stance the  real-base  will  be  the  5th  of  the  discord.  This  tone  is  to 
be  omitted  above.  The  upper  parts  of  the  last  example  should  be 
re-arranged  in  two  other  positions,  without  altering  the  base  part. 

A  modulatory  theme,  with  the  treble  parts  added,  is  here  pre- 
sented to  illustrate  this  theory.  The  student  should  write  the  base 
part.  Every  discord  except  the  last  appears  in  its  second  inversion  : 

Ex.  248. 

'   I^BZFf^r*; 

feZ      ': 


Only  such  places  as  are  marked  -f  are  to  be  real-bases ;  otherwise 
the  base  is  to  have  the  root  of  each  chord.  The  solution  of  this  will 
be  included  in  the  Key.  The  example  should  be  re-arranged  and 
transposed  into  several  keys.  *  *  * 


112 


GOODRICH'S  ANALYTICAL  HARMON \. 


When  the  modulations  are  to  the  3d  above,  the  same  principles 
may  be  applied ;  but  in  these  instances  the  real-base  will  be  the  3d 
not  the  5th  of  the  discord.  No  connecting  note  appears  between  the 
discord  and  its  antecedent ;  but  when  the  modulation  is  from  niiuo] 
to  relative  major,  the  chord  progression  is  easily  managed : 


Ex.  249. 


(i) 


The  parallel  fifths,  c  —  g,  d  —  a-flat,  are  allowable,  because  the  firs 
fifth  is  normal  and  the  second  is  imperfect.  The  reverse  of  thi: 
order  is  not  good.  What  adds  most  to  the  effectiveness  of  the  firs 
progression  is  the  contrary  movement  of  the  octave,  c  to  d,  below 
and  c  to  b-flat  above.  The  resolution  of  the  discord  is  perfect!] 
regular.  The  figure  i  refers  to  the  first  inversion,  the  3d  of  the  dis 
cord  being  below,  as  real-base. 

This  example  is  shown  in  three  positions,  that  the  student  ma} 
observe  its  different  phases.  The  3d  and  5th  of  the  triad  ascend  will 
the  base,  while  the  duplicated  root-note  above  descends  a  whole  step 
The  real-base  must  be  a  minor  2d  below  the  resulting  tonic,  and  ii 
some  places  a  chromatic  alteration  must  be  supplied  by  the  student 

A  theme  and  upper  parts  are  presented  as  illustrations.  Tin 
student  is  to  add  the  base  part  according  to  the  same  principles  tha 
governed  the  previous  example. 

The  real-bases  are  indicated.  In  these  places  first  ascertain  thi 
root  of  the  discord,  then  write  the  3d  in  the  base. 

The  note  omitted  from  the  treble  part  is  thus  supplied  by  th< 
base: 


Ex.  250. 


Re-arrange  the  treble  parts  in  two  other  positions.  Transpos( 
to  C-minor,  D-minor,  and  F-sharp  minor. 

One  or  two  examples  should  also  be  worked  out  from  the  base 
Such  ground-work  is  here  transcribed  for  the  student  to  build  upon 


GOODRICH  S   ANALYTICAL    HARMONY. 
4-  +7         Skip.        4- 


"3 


II)  (1)  (1) 

The  notes  marked  7  are  tc  oe  roots  of  dominant  yth  chords.  (1} 
signifies  the  first  inversion  of  a  dominant  jth  chord,  the  3d  being  in 
the  base. 

The  following  exercise,  if  worked  out  in  various  keys  and 
positions,  will  be  found  useful : 


Ex.  252, 


;fe?— * — 0      t      ^T=^=l 


(1) 


-* 0- 


Chapter  XXVIII. 


UNRULABLE  PROGRESSIONS  AND  RESOLUTIONS. 

/HpHERE  are  so  many  seeming  contradictions  to  the  rules  of  com- 
•*•  position  that  the  author  of  this  system  has  set  forth  as  few  as 
possible.  Musical  rule  can  exist  only  as  a  deduction  from  musical 
usage.  The  creative  artists  are  the  highest  authority  respecting  the 
material  of  composition  and  its  application.  If  Beethoven  caused  a 
yth  to  ascend,  it  is  the  duty  of  the  theorist  to  show  why  the  com- 
poser did  so ;  not  to  stand  aloof  and  shake  his  head  with  the  remark, 
"this  is  a  violation  of  our  rules  ! "  This  has  always  been  the  custom. 
Yet  who  made  these  rules?  Were  they  engrossed  and  put  forth  by 
some  one  greater  than  Beethoven  ? 

Directions  concerning  the  resolution  of  a  discord  can  not  be 
given  until  it  is  known  what  application  is  to  be  made  of  this  dis- 
cord. When  the  chord  that  is  to  follow  the  discord  is  determined 
upon,  as  well  as  the  situation  in  which  both  occur,  then  may  certain 
precepts  be  followed  to  advantage.  But  there  is  an  underlying 
principle  that  affords  the  solution  of  every  musical  problem,  and  that 
is :  the  object  in  view,  or  the  esthetic  effect  desired. 


GOODRICH'S   ANALYTICAL    HARMONS 


Music  students  should  endeavor  to  grasp  these  principles  and 
apply  them ;  not  to  imagine  that  the  memorizing  of  rules  and  for- 
mulas will  be  sufficient. 

The  example  illustrates  a  connecting  note  passing  into  another 
voice-part  instead  of  remaining  stationary : 

Raff. 


Ex.  253. 


.^; 


•j-j-^Tteq 


B^ 


This  does  not  agree  with  the  connecting-note  theory,  but  it  is  un- 
avoidable here.  All  that  can  be  said  against  the  progressions  in- 
dicated thus  ^  is,  that  they  are  not  smooth  and  connected.  It  is 
unprofitable  to  look  upon  them  as  .contradictions  of  a  rule,  especially 
since  no  error  results.  Though  such  progressions  have  been  fre- 
quently used  to  advantage,  the  student  must  not  conclude  that  the 
connecting-note  principle  is  to  be  dispensed  with. 

In  the  great  majority  of  progressions  the  previqus  directions  will 
be  applicable,  and  should  be  followed.  But  in  harmonizing  a  theme 
ascending  from  the  tonic,  the  most  available  method  is  that  employed 
in  the  last  example,  in  which  the  base  moves  contrarily  to  the  other 
parts.  If,  however,  smoothness  and  connection  were  desirable,  the 
melody  could  be  harmonized  in  this  manner : 


Ex.  254. 


There  is  a  connecting  note  throughout,  in  the  mezzo-soprano  part, 
and  the  base  moves  alphabetically.  Each  example  serves  a  particu- 
lar purpose,  and  the  purpose  must  justify  the  method  employed. 

Here  are  other  progressions  of  a  character  similar  to  those  in  Ex. 
253: 

&=*=l=tA 


Ex.  255. 


* 


*=* 


-: 


& 


GOODRICH 'S   ANALYTICAL    HARMONY. 


At  (a)  three  parts  skip,  while  the  mezzo-soprano  part  moves  alpha- 
betically. The  same  is  true  of  (b).  In  neither  case  does  the  con- 
necting note  remain  in  the  same  part.  It  would  be  childish  to  con- 
demn these  progressions  merely  because  they  do  not  comply  with 
the  directions  as  to  chord  succession  in  general.  At  (c)  all  the  parts 
leap  a  considerable  distance.  But  the  only  change  in  harmony  con- 
sists in  the  introduction  of  the  yth.  Besides,  the  parts  move  in- 
opposite  directions.  All  these  progressions  are  correct,  though  some- 
what irregular. 

The  contrary  resolution  (progression)  of  the  3d  and  yth  of  a  dom- 
inant yth  chord  is  now  in  order.  Take  the  3d  first.  The  rule  is 
that  it  must  ascend  a  minor  zd  in  going  to  the  tonic  chord.  It  may 
also  descend  a  $d. 

In  the  middle  of  a  strain,  or  wherever  it  is  not  desirable  to  make 
an  authentic  cadence,  the  3d  may  descend  to  the  5th  of  the  tonic 
chord : 


Ex.  256. 


The  author's  explanation  of  these  seeming  contradictions  of  musical 
rule  is,  that  they  are  Progressions  not  Resolutions ;  that  they  occur 
in  passages  where  the  decisive,  compulsory  character  of  a  direct 
resolution  is  not  desirable,  and  that  in  such  places  they  serve  a  dis- 
tinct purpose.  This  must  be  understood  as  an  intermediate,  not  a 
final  application.  The  next  example  illustrates  this : 


/_        ! 

,         • 

2-K    * 

A*_ 

@fr—  I 

-  r^« 

i  — 

-p~"»— 

1 

L-*  —  • 

^  i* 

^~ 

1  — 

— 

=i  —  I—H 

* 

<i 

* 

£2=*=^= 


Ex.  257. 


In  the  first  progression  the  3d  of  the  discord  skips  down  to  g,  but 
in  the  last  cadence  this  leading-note  resolves  regularly  and  naturally 
up  to  the  tonic. 

A  similar  liberty  may  be  taken  with  the  yth  of  tha  discord. 
According  to  rule  it  must  descend  a  2d;  it  may  also  ascend  a  2d, 
provided  it  is  below  the  3d,  and  that  it  occurs  intermediately : 


n6 


GOODRICH  S   ANALYTICAL    HARMONY. 


Ex.  258. 


At  (a)  there  is  a  false  progression  by  fifths,  in  addition  to  the  irreg- 
ular progression  of  the  yth  up  to  the  5th  of  the  concord.  It  should 
therefore  be  condemned.  The  parallel  fifths  are  avoided  at  (b)  and 
(c).  The  3d  ascends  to  the  tonic,  and  these  two  progressions  are 
allowable  in  an  intervening  passage,  notwithstanding  the  upward 
movement  of  the  yth.  The  effect  of  these  progressions  is  somewhat 
similar  to  that  of  the  third  resolution  of  the  dominant  yth  chord: 
they  are  both  undecided,  and  lead  us  to  expect  something  else.  For 
these  reasons  the  upward  movement  of  the  yth  may  be  excellent, 
and  in  place  of  condemning  such  progressions  they  may  be  entitled 
to  more  praise  than  a  regular  resolution.  This  presupposes  that 
they  occur  elsewhere  than  in  a  final  cadence,  and  that  progression, 
rather  than  resolution,  is  the  evident  intention. 

An  example  is  now  presented  wherein  these  theories  are  illustrat- 
ed by  means  of  notation  : 

B.  Godard. 


Ex.  259 


--I-    -£— •* — *5» 


m 


During  the  first  full  measure  the  yth  (e-flaf)  ascends  to  the  5th  of 
the  concord  in  order  to  preserve  the  harmonic  sequence. 

These  are  mere  progressions  occurring  in  the  middle  of  a  phrase 
where  it  is  not  desirable  to  resolve  the  discord  decidedly.  But  at  the 
close  the  jth  descends  and  the  3d  ascends  in  regular  order,  leaving 
nothing  to  be  desired,  so  far  as  harmonic  completeness  is  considered. 
This  example  does  not  violate  the  rules,  as  theorists  have  supposed  ; 
it  merely  serves  to  demonstrate  the  particular  application  of  rides. 

When  the  3d  of  the  discord  descends  the  7th  should  be  resolved 
rulably ;  when  the  yth  ascends  the  3d  should  resolve  up  a  half  step. 
/Or,  the  3d  may  be  omitted  when  the  yth  ascends.  This  will  prevent 
parallel  fifths,  thus : 


GOODRICH'S  ANALYTICAL  HARMONY. 

Fesca. 


Ex.  260. 


Ho  objection  ought  to  be  raised  against  this,  for  it  is  excellent.  An 
t'xample  from  Beethoven  is  here  quoted  as  a  farther  illustration  of 
these  irregular  resolutions : 

Op.  28. 

u 


Ex.  261. 


In  both  instances  the  yth  (e)  ascends.  But  this  occurs  in  an  inter- 
mediate progression,  not  in  a  final  resolution.  Hundreds  of  similar 
examples  might  be  quoted. 

There  is  not  much  exercise  work  to  be  performed  in  connection 
with  this  chapter,  but  the  student  should  read  it  by  paragraphs  until 
the  substance  is  mentally  digested.  Each  example  might  be  tran- 
scribed into  other  keys  as  a  farther  means  of  reducing  the  theories 
to  practical  operation. 


$v$~£2  

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y-     'JH  ' 

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,, 

t-\.>    "^i  —  p  ^  — 

>» 

3 

fi             *^       i 

-•*•  «  —  j 

Chapter  XXIX. 


DISSONANT  TRIADS— IMPERFECT,  AUGMENTED 
AND  DIMINISHED. 

THE  IMPERFECT   TRIAD. 

OpHOUGH  the  imperfect  triad  was  excluded  from  the  preliminary 
-*-    examples,  its  intervals  are  included  in  the  essential  yth  chord. 
Consequently  but  little  remains  to  be  explained. 


ti8  GOODRICH'S  ANALYTICAL  HARMONY. 

This  triad  is  founded  upon  the  leading-tone  of  every  major  scale, 
and  upon  the  super-tonic  and  leading-tone  of  every  minor  scale.  It 
consists  of  two  minor  thirds,  component!?  ;  or  of  a  minor  3d  and 

imperfect  5th  :  '  ^.^~^^^          ]  Not  being  a  concord,  it  must 

be  classified   among  discords,  and  some  general  directions  will   be 
given  for  its  resolutions. 

If  it  be  considered  as  belonging  to  tonic  major,  the  root  should 


ascend  and  the  5th  descend  :  Ex.  263.         z=^=      These  elements  of 


transition  correspond  to  the  3d  and   jth  of  the  essential  discord, 


which  disappear  in  the  same  manner  :   Ex.  264.  Fij^^iib:       The  3d 


usually  descends  a  2d.  These  are  the  mobt  natural  tendencies  of 
the  different  tones  toward  resolution,  though  the  triad  has  been 
used  in  various  ways. 

A  few  illustrations  in  three-part  harmony  are  presented  as  the 
imperfect  triad  usually  appears  in  this  manner  : 


j— 

I     I     I     I       I  |       I  I 

In  whatever  position  the  dissonant  triad  may  be  placed  it  can  be 
resolved  in  this  manner  without  fear  of  impropriety.  This  is  per- 
haps the  best  reason  for  following  the  above  plan.  The  imperfect 
triad  is  indicated  by  a  cross. 

By  assuming  a  melodic  license  the  3d  may  ascend  a  4th  or  de- 
scend a  5th,  provided  the  two  elements  of  transition  (root  and  5th) 
resolve  regularly  : 

I 


Ex.  266. 


*=f^^= 

i3z^^-d 


This  supplies  the  remaining  note  of  the  tonic  chord  and  is  correct. 
Transpose  each  of  these  examples. 

In  four-part  harmony  the  root  or  3d  may  be  used  as  base,  and 
these  notes  may  also  be  doubled — always  provided  that  no  improper 
progressions  result.  The  5th  is  seldom  used  as  a  real-base.  An 
exception  may  be  observed  in  Ex.  245. 


GOODRICH'S  ANALYTICAL  HARMONY. 


119 


In  associating  this  triad  with  the  minor  of  the  tonic,  the  same 
observations  will  apply : 

_,J^  J      *k  I 
Ex.  267.  ^^^^f 


There  is  nothing  new  here,  except  the  difference  in  mode.  But 
when  this  imperfect  triad  is  used  in  connection  with  A-minor  the 
treatment  should  be  different,  as  the  tonic  will  be  below  the  root.  If 
this  root-note  in  the  base  is  moved  down  to  the  tonic,  parallel  5ths 
will  invariably  result. 


Ex.  268. 


If  the  root  be  resolved  up  to  C,  the  final  chord  will  be  recognized  as 
that  of  C-major,  not  A-minor.  The  only  remaining  choice  would 
be  to  lead  the  5th  (/)  up  a  major  3d  to  the  minor  key-tone ;  for  the 
5th  of  a  concord  may  be  dispensed  with,  but  a  root  can  not  be 
omitted  without  establishing  in  its  place  some  other  root.  The 
example  would  appear  like  this : 


Ex.  269. 


These  examples  are  sufficiently  correct,  but  they  partake  of  the 
nature  of  progression,  rather  than  of  resolution.  The  natural  ten- 
dency of  the  f  is  to  descend  a  minor  2d  to  e,  and  the  skip  up  to  a  is 
therefore  an  expediency.  But  the  imperfect  triad  founded  on  the 
leading-tone  to  A-minor  corresponds  exactly  to  that  one  used  in  Ex. 
267.  With  this,  no  difficulty  will  be  experienced,  for  there  is  here  a 
note  that  resolves  naturally  to  the  tonic,  and  the  other  intervals 
disappear  according  to  their  melodic  tendency  without  endangering 
the  smoothness  and  correctness  of  the  \vhole  resolution  : 


Ex.  270. 


I2O 


GOODRICH'S  ANAIA  TICAL  HARMONY. 


Before  dismissing  this  somewhat  ambiguous  triad,  a  few  instances 
will  be  recorded  in  which  it  is  treated  in  chord  progression  the  same 
as  a  concord.  That  is,  it  may  supply  an  accompaniment  to  that  tone 
of  the  scale  upon  which  it  is  founded : 


Ex.  271. 


In  each  of  the  first  three  measures  the  imperfect  triad  takes  its  place 
among  the  perfect  triads  in  order  to  carry  out  the  sequence*  indicat- 
ed by  the  slurs.  The  fact  that  the  dissonant  triad  ascends  and  de* 
scends  without  any  fixed  resolution  shows  that  it  assumes  here  the 
functions  of  a  concord,  wrhich,  of  course,  has  no  resolution.  But  at 
the  close,  after  the  sequence  has  been  carried  out,  the  dissonant  triad 
is  resolved  as  though  its  tones  belonged  to  the  essential  jth  on  C. 
The  following  extract  from  a  favorite  English  song  illustrates  the 
same  theory  in  a  different  manner : 


fiishoft. 


In  the  second  and  fourth  measures  of  the  accompaniment  the  im- 
perfect triad  is  treated  like  the  concords,  these  being  progressions, 
not  resolutions.  Attention  is  also  directed  to  the  manner  in  which 
parallel  fifths  are  avoided.  The  5th  between  the  base  and  lower 
treble  part  becomes  a  6th  before  the  base  ascends  from  c.  Then 
the  c  above,  which  now  produces  a  5th  with  the  f-sharp,  ascends 
to  d.  By  moving  these  two  parts  alternately,  false  progressions  are 
avoided. 

*  Sequence  is  the  repetition  upon  different  degrees  of  the  scale  of  anv  figure  or  design 
considered  as  a  model.  The  upper  part  during  the  first  three  measures  constitutes  a  rielodic 
sequence;  the  chords  all  being  in  the  same  position,  constitute  what  is  nere  caUed  Har- 
monic Sequence. 


GOODRICH'S  ANALYTICAL  HARMONY. 


121 


The  impel  feet  triad   is  frequently  used   in  a  minor  cadence  in 
place  of  the  sub-dominant  harmony  : 

Haendel. 
A Ur-t- 


Ex.  273 


(2)     (2) 


The  3d  is  doubled  to  avoid  similar  movement  in  the  upper  parts. 
Observe  that  the  root  (b)  may  descend  when  it  appears  above  the 
5th.  The  usual  method  is  to  use  the  3d  as  a  real-base : 


J'.E.  Bach. 


Ex.  274. 


(1)     (2) 


The  imperfect  triad  becomes  a  sub-domimant  harmony  in  this  in- 
stance, b  being  substituted  for  a  on  account  of  the  melody.  The 
real-base  is  doubled  above,  but  one  d  ascends  to  e  while  the  other 
descends  to  c.  The  last  example  may  be  written  in  this  form : 


Ex.  275. 


P= 


m 


Care  must  be  bestowed  upon  the  re-arrangement  of  these  dissonant 
triads.     Transpose  the  last  three  examples  into  several  keys. 

THE  AUGMENTED  TRIAD. 

The  word  augmented,  as  applied  to  intervals,  refers  to  the  en- 
largement of  a  major  or  normal  interval  by  one  chromatic  tone. 
This  may  be  accomplished  by  sharpening  the  upper  tone  of  the  in- 
terval or  flattening  the  lower  tone.  The  former  process  is  more 
common. 

The  3d  or  5th  of  a  maj  or  chord  may  be  augmented,  but  in  this 
chapter  the  augmented  5th  only  will  be  considered.  Select  any 
major  chord,  and  by  sharpening  the  5th  an  augmented  triad  will 


122 


GOODRICH'S  ANALYTICAL  HARMONY. 


result :  Ex-  2?6- 


The  nature  of  this  chord,  containing 


as  it  does  two  major  thirds,  is  harsh,  and  it  has  a  strong  tendency 
towards  immediate  resolution.  The  object  of  d-sharp  is  to  ascend 
to  e.  The  other  two  notes  may  remain,  or  the  upper  major  3d  may 
ascend  to  c  and  e,  while  the  root  remains  as  connecting  note.  Tlu- 
two  examples  follow : 


Ex.  277. 


The  second  of  these  resolutions  is  of  more  frequent  occurrence,, 
though  both  are  useful.  The  base  to  these  exercises  is  easily  man- 
aged. In  either  instance  it  may  proceed  from  root  to  root,  as  though 
the  augmented  interval  did  not  appear : 


Ex.  278. 


-*1 


5hK 


t=g 


Arrange  this  in  two  other  positions  and  transpose.  (There  is  another 
mode  of  treating  the  base,  but  it  can  not  properly  be  introduced 
here.) 

The  augmented  triad,  on  account  of  its  dissonant  character,  re- 
quires preparation.  But  as  it  is  already  prepared  in  the  examples. 
any  farther  explanation  of  this  subject  may  be  left  to  a  future 
chapter.  Frequently  the  3d  of  the  augmented  triad  is  used  as  a 
real-base,  that  tone  being  omitted  above : 

J* 


Ex.  279. 


In  four-part  harmony  the  root,  5th  or  base  may  be  doubled ;  making 
two  treble  and  two  base  parts.  Rubinstein  has  even  given  the 
augmented  5th  to  the  base  and  with  charming  effect,  as  this  quota- 
tion will  prove : 


GOODRICH'S  ANALYTICAL  HARMONY. 


123 


BALLET    MUSIC    FROM  "  FERAMORS." 


Ex.  280. 


The  augmented  5th  is  included  in  the  middle  part  in  order  to  pre- 
serve this  design  in  the  accompaniment : 


Ex. 


The  full  effect  of  the  augmented  triad  on  the  second  beat  of  meas- 
ures i  and  3  is  also  more  satisfactory  and  complete  than  if  the  upper 
d-sharp  had  been  omitted. 

THE  DIMINISHED  TRIAD. 

Theorists  are  agreed  that  a  diminished  interval  is  one  chromatic 
step  smaller  than  a  minor  interval.     Therefore  diminished  presup- 

poses minor,  but  with  regard   to  this  interval  :  Ex'  282- 

which  is  generally  called,  a  diminished  5th,  there  is  a  contradiction 
to  be  noted.  By  enlarging  this  so-called  diminished  5th  one  chro- 
matic step  the  result  should  be  a  minor  5th  in  order  to  make  the 


theory  consistent.      But  these  fifths :    Ex-  283-  Ffen 


have 


never  been  called  minor.  The  old  theorists  called  them  "perfect," 
though  in  strict  designation  they  are  not  absolutely  pure.  For  this 
reason  Weitzmann  terms  them  "  major  fifths."  Riemann  says  they 
are  "  standard  fifths."  But  since  these  intervals  are  the  same  in  both 
modes,  as  well  as  by  inversion,  the  author  has  applied  the  term  Nor- 
mal to  the  4th  and  5th  of  every  normal  scale.  For  this  interval : 


Ex.  284. 


the  most  appropriate  name  seems  to  be    Imperfect. 


It  might  be  called  minor,  but  it  can  not  consistently  be  called  dimin- 
ished.    By  flattening  the  /  or  sharpening  the  b,  a  diminished  5th 


124 


GOODRICH 'S    ANALYTICAL    HARMONY. 


will  result.     This  interval  has  been  used  and  will  be  included  here- 
after among  altered  intervals. 

The  same  process  is  carried  out  in  forming  a  diminished  yth  or 
a  diminished  3d,  thus  : 

Mnj.         Mill.         Dim. 


Ex.  285. 


w 

Min. 


LX> 1 


Dim. 


•& — P-frs*— — \?s> — 


W 

The  diminished  triad  is  now  resumed.     This  will  be  formed  from  a 
minor  triad  by  raising  the  root :  Ex-  a86'  \3gEE£E$sr:E:{   From  «-j//a;^ 


to  c  is  a  diminished  3d,  and  from  this  the  triad  is  called  diminished. 
In  this  position  the  parts  are  brought  so  near  together  that  it  is  not 
favorable  to  practical  application.  The  root,  or  the  middle  note, 
may  be  inverted,  thus  securing  a  better  position.  With  regard  to 
the  resolution,  two  of  the  notes  have  a  fixed  progression  to  the 
unison  b : 


The  5th  of  the  triad  may  descend  a  whole  or  a  half  step  .• 


Ex.  288. 


The  former  is  more  unusual,  and  for  this  reason  seems  less  satisfac- 
tory. However,  as  it  is  sufficiently  correct  it  may  serve  a  purpose. 

Transpose  the  last  example,  using  both  resolutions. 

A  diminished  triad  may  also  be  produced  by  raising  or  lowering 
the  3d  of  an  imperfect  triad.  In  the  first  instance  the  upper  3d  ap- 
pears diminished;  in  the  second  instance  the  lower  3d  is  diminished." 


Ex.  289. 


GGODRiCII'3    ANALYTICAL    HARMONY.  12.5 

An  open  position  is  more  lavorabie  tor  tnese  discords,  on  account 
of  the  ascending  and  descending  tendency  of  the  extreme  parts : 


A    A.      f 


Ex.  290. 


At  (a)  the  3d  is  placed  below,  as  a  real-base ;  at  (b)  the  root  remains 
below.  The  latter  is  more  satisfactory,  and  accordingly  more  use- 
ful. The  resolution  at  (a-)  is  so  ambiguous  as  to  suggest  some  con- 
tinuation beyond  this  point,  as  : 


Ex.  291. 


The  discord  is,  therefore,  an  intermediate  one,  and  not  suited  to  a 
final  close.  The  same  may  be  said  of  the  discord  at  (b)  and  its 
resolution,  though  this  is  more  satisfactory,  at  least  in  a  mere  theo- 
retical exercise,  where  the  ulterior  design  does  not  appear. 

Transpose  the  last  two  examples.     The  latter  should  be  contin- 
ued to  a  satisfactory  close  by  adding  two  or  three  measures. 


126  GOODRICH'S  ANALYTICAL  HARMONY. 


PART  VII. 


Chapter  XXX. 


ORIGIN  AND  PRINCIPAL  RESOLUTION  OF  THE 
DIMINISHED  SEVENTH  CHORD. 

BY  sharpening  the  root  of  any  dominant  7th  chord  there  results 
a  diminished  yth  chord;  as  any  minor  interval  lessened  by  one 

M  x^j^"^*"x^ 

chromatic  step  becomes  diminished  :  E>x-  292-  NK^EjiizjJazE      From 

the  root  to  the  highest  note  of  the  first  discord  (a  to  g)  is  a  minor 
7th;  a-sharp  lessens  the  interval  and  makes  it  diminished.  The  3d, 
5th,  and  7th  of  the  dominant  chord  remain  stationary  as  parts  of  the 
next  discord,  while  the  base  is  raised  one  chromatic  step. 

The  chromatic  alteration  of  the  first  discord  changes  its  name, 
nature,  and  resolution,  and  results  in  a  principal  diminished  jth 
chord  upon  A-sharp.  The  latter  consists  of  a  minor  3d,  imperfect 
5th  and  diminished  7th;  or,  a  compound  of  three  minor  thirds. 

The  first  discord  belongs  to  D-major,  the  second  to  B-minor: 

E,29,Ffe%=iz3ER|^ 


I>-maj.  B-min. 

The  roots  of  the  two  discords  represent  the  difference  between  the 
scales  of  the  two  modes,  D-major  and  R-minor. 

The  natural  resolution  of  the  diminished  jth  chord  is  to  the 
minor  concord  founded  a  minor  2d  above  the  root  of  the  discord. 
The  root  of  the  diminished  jth  chord  is  the  leading-tone  of  the  key 
to  which  it  naturally  resolves.  This  is  one  of  the  strongest  argu- 
ments in  favor  of  the  harmonic  minor  scale,  as  a  characteristic  series 


GOODRICH'S  ANALYTICAL  HARMONY. 


o;  tones ;  for  the  diminished  ~th  chord  and  its  principal  resolution 
comprise  every  tone  in  the  scale,  tnus : 
,    i  3 

i — IT 

Ex.  294 


^ 


•^K=r5&=zgs 

-"-— -z=bg:g=g:? 


As  the  diminished  chord  occurs  in  this  scale  only,  it  may  be  con-( 
eluded  that  it  belongs  naturally  to  A-minor.  Observe  that  the  dimin- 
ished yth  chord  contains  both  elements  of  transition  (leading-note 
and  sub-dominant)  to  A-minor. 

The  student  should  now  write  a  diminished  yth  chord  in  each  of 
the  fifteen  keys.  In  every  instance  the  diminished  yth  is  to  be  de- 
rived from  the  dominant  yth  chord.  Write  the  signature  of  each 
key  in  regular  order,  and  locate  the  dominant  yth  discord  upon  the 


5th  degree  of  the  major  scale:    Ex>  295'  tfezz  ^  — 


Each 

example  is  to  begin  in  major  and  terminate  in  the  relative  minor. 
The  minor  concords  are  to  appear  in  their  first  position. 
The  next  example  in  regular  order  follows  : 

Ex-  2Q6-  E-£=i—  jjjjj==§=| 

L        [      -H[     -L-^=3 

Proceed  with  the  remainder  as  directed.  When  the  root  of  the  first 
discord  is  sharpened  by  the  signature  it  will  be  necessary  to  use  a 
double  sharp  for  the  root  of  the  second  discord,  example  : 

.    ..J  -. 
Ex.  297. 

The  example  in  C-flat  will  correspond  to  the  one  beginning  in  Pi- 
rn a  j  or  :  Ex.  298. 


<%^       These    are    enharmonic 


equivalents.     * 

This  is  not  the  only  derivation  of  the  chord.  It  is  also  a  product 
of  the  harmonic  minor  scale,  and  may  be  entirely  independent  of  an 
essential- yth  chord.  This  will  appear  hereafter. 

The  principal  resolution  of  the  diminished  yth  chord  will  now  be 
considered.  The  root,  being  the  leading-tone  to  the  key  in  which  it 
belongs,  ascends  a  minor  2d  ;  the  jth  must  descend  a  minor  2d  <  be- 
cause there  is  no  other  interval  of  the  tonic  triad  to  which  it  will 


128 


GOODRICH'S  ANALYTICAL  HARMONY. 


resolve) ;  the  5th  of  the  diminished  chord  is  the  sub-dominant,  &n< 
must  descend  a  major  2u. 

Here  are  the  three  most  important  notes  :  Ex.  299.  EJE— ^: 

The  5th  of  the  diminished  yth  chord  must,  accordingly,  be  treate< 
the  same  as  the  dominant  7th  to  the  same  key,  because  this  note  (c 
here)  has  the  same  relationship  in  both  chords: 


Ex.  300.  H£r— g""^ 


The  white  notes  show  the  two  elements  of  transition,  and  their  reso 
lution,  first  in  the  diminished,  and  then  in  the  corresponding  dom 
inant  yth  chords.  The  black  notes  show  that  while  the  7th  of  tin 
diminished  chord  descends  to  the  5th  of  the  concord,  the  root  of  tin 
dominant  jth  chord  remains  stationary.  (Compare  (a)  with  (b),  Ex 
300.)  The  leading-note  at  (a)  is  the  root  of  the  discord  ;  at  (b)  it  i; 
the  3d  of  the  discord ;  but  as  both  are  transition  chords  to  A,  botl 
elements  of  transition  must  be  treated  alike.  The  3d  of  the  dimin 
ished  7th  chord,  like  the  5th  of  the  dominant  7th  chord,  has  no  de 
cided,  fixed  resolution,  but  may  ascend  or  descend,  according  to  th< 
position  of  the  chord  : 


Ex.  301. 


*&— f—\ rfe*=g= 


The  intervals  which  have  a  fixed  resolution  are  indicated  in  Ex 
299.  By  analyzing  the  discord  it  will  be  found  to  contain  two  in 
tervals  of  a  5th,  from  i  to  5  and  3  to  7  : 


Ex.  302. 


Consequently  if  either  of  these  fifths  be  resolved  in  similar  move 
ment  parallel  fifths  will  result,  as  in  the  second  measure.     There 


GOODRICH'S  ANALYTICAL  HARMONY. 


129 


fore,  whenever  the  fifths  appear,  they  must  be  resolved  in  contrary 
movement,  as  here : 


Ex.  303. 


This  is  correct.  The  parts  that  move  in  similar  directions  are  sit- 
uated a  loth  from  each  other,  thus :  i  and  3,  5  and  7.  By  resolving 
the  principal  intervals  of  the  discord  (root,  5th  and  7th)  as  directed, 
no  raise  progressions  can  result  in  any  position.  But  when  the  7th 
is  uppermost  the  3d  must  ascend.  When  the  3d  or  5th  is  upper- 
most the  3d  may  (and  generally  does)  descend  to  the  tonic.  The 
two  positions  most  available  for  present  purposes  are  these : 


Ex.  304. 


~ 


The  upper  5th  (as  f-sharp  and  c]  appears  in  both  measures  as  a  4th. 
Therefore  the  3d  or  5th  of  the  diminished  chord  is  to  be  in  the 
melody  whenever  the  chord  is  used  in  its  principal  resolution. 

It  is  not  advisable  at  present  to  invert  the  base.  As  a  prepara- 
tion for  the  following  chapter,  transpose  the  last  example  into 
several  keys. 


GOODKICII  S    ANALYTICAL    HARMONY. 


Chapter  XXXI. 


NATURAL  MODULATIONS  TO  RELATED  MINOR 

KEYS  BY  MEANS  OF  THE  DIMINISHED 

SEVENTH  CHORD. 

ANOTHER  MODE  OF  TRANSITION. 

THE  chord  of  the  diminished  yth  affords  another  means  o)  tran- 
sition to  the  related  minor  keys.     The  classification  of  these 
here  follows : 

Ex.  305. 

I  a  -i 


The  upper  staff  here  contains  the  three  dominant  yth  chords  to  the 
three  major  triads ;  the  lower  staff  contains  the  three  diminished  y  th 
chords  to  the  three  minor  triads.  Each  minor  triad  below  is  the 
relative  of  the  major  triad  above. 

Compare  the  exercises  vertically  :  i  (a)  with  i  (b),  2  with  2,  and 
3  with  3.  Each  diminished  yth  chord  is  the  natural  consequent  of 
the  antecedent  dominant  yth  chord  above  it. 

The  three  diminished  yth  chords  representing  the  three  related 
minor  keys  to  the  normal  scale  of  C  are,  strictly  speaking,  the  only 
diminished  yth  chords  in  music,  all  others  being  mere  enharmonic 
alterations  of  these  primary  chords.  Consequently  it  is  important 
that  the  pupil  should  acquire  thorough  control  of  these  discords. 

As  already  ascertained,  the  positions  of  a  diminished  chord  best 
adapted  to  present  requirements  are  those  in  which  the  3d  or  5th  is 
uppermost.  In  employing  the  diminished  yth  chords  in  the  follow- 
ing harmonization  introduce  the  antecedent  dominant  yth  chord 
before  each  diminished  chord. 

The  difference  between  the  two  discords  is  in  the  roots ;  the  3d, 
5th  and  yth  remaining  the  same. 


GOOI'RICH'S    ANALYTICAL    HARMONY. 


This  melody  contains  transitions  by  means  of  diminished  7th 
chords  to  the  three  related  minor  keys : 
Ex.  306. 


The  first  chord  may  be  either  C-major  or  E-minor,  the  second  must 
be  a  dominant  yth,  the  third  must  be  the  consequent  diminished  jth, 
and  the  fourth  chord  must  be  a  minor  triad  founded  a  minor  2d 
above  the  root  of  the  diminished  7th  chord.  The  dashes  show  where 
the  resolutions  of  the  diminished  7th  chords  take  place.  If  the 
student  requires  a  chart  for  the  harmonization  of  this  theme,  Ex.  305 
may  be  consulted.  It  would  be  well,  however,  to  write  without  a 
chart,  if  possible. 

The  repeated  notes  are  to  be  considered  as  connecting  notes. 

%.     %     ^c  • 

The  only  unusual  progression  in  the  harmonization  occurs  in 
the  third  measure.  There  is  but  one  concord  which  can  properly 
accompany  the  first  g  of  the  third  measure.  From  this  triad  to  the 
following  dominant  7th  chord  involves  a  progression  not  heretofore 
employed, — though  there  are  two  connecting  notes. 

As  the  base  ascends  a  second  in  all  the  resolutions  of  the  dimin- 
ished 7th  chords,  the  treble  parts  must  all  descend.  No  connecting 
note  appears  in  the  principal  resolution  of  a  diminished  7th  chord, 
therefore  contrary  movement  is  in  keeping  with  previous  directions. 
(This  is  not  to  be  re-arranged.)  Transpose  into  G,  B-flat  and  A-flat. 

An  illustration  of  the  facts  to  be  comprehended  appears  in  the 
following  chart  in  A-flat  : 

Ex.  307. 


Dom.  7th. 


Dim.  7th. 


Minor  triads. 


Major  triads. 

The  dominant  7th  chord  marked  i  represents  the  key  of  the  major 
triad  with  a  corresponding  number.  So  with  2  and  3.  The  dimin- 
ished 7th  chords  numbered  i,  2,  3  represent  the  same  keys  as  the 
minor  triads  whose  numbers  ar2  the  same.  Also,  the  antecedent  of 
each  diminished  chord  is  its  corresponding  number  among  the  dom- 
inant 7th  chords,  i  among  the  diminished  chords  is  derived  from  r 
among  the  dominant  7th  chords.  The  numbers  and  the  chords 
correspond  in  four  different  ways,  the  last  of  which  will  be  men- 


132 


GOODRICH'S   ANALYTICAL    HARMONY. 


tioned  :  The  minor  triad  marked  i,  is  the  relative  of  the  major  triad 
marked  i ;  and  the  same  may  be  said  of  the  other  triads  :  their  num- 
bers correspond  to  each  other. 

Examine  Ex.  307  attentively,  and  re-read  the  explanations  follow- 
ing the  example.  Also  transpose  the  chart  into  several  keys. 

One  other  simple  theme  is  presented  in  order  to  illustrate  another 
position  of  the  diminished  chords : 

Ex.  308. 


The  dashes  indicate  the  resolution  of  the  diminished  yth  chords. 
Begin  and  end  in  C-major.  Transpose  to  D  and  E.  * 

The  next  exercise  (mostly  in  minor)  introduces  the  diminished 
chord  without  its  antecedent  dominant  yth.  The  diminished  chords 
are  here  introduced  independently,  but  their  resolution  remains  the 
same. 

The  student  should  complete  the  harmony  by  supplying  the  base : 

Ex.  309. 


The  dashes  indicate  the  three  diminished  chords  resolving  to  the 
three  minor  chords  in  this  key.  Real-bases  are  to  be  included  at  (2) 
and  (i).  The  first  chord  in  the  fifth  measure  is  to  be  an  imperfect 
triad  with  the  3d  in  the  base.  During  the  last  of  this  measure  an 
avoided  cadence  is  outlined.  This  postpones  the  final  cadence  until 
the  very  last  of  the  example. 

Transpose  the  theme  alone  into  C-minor  and  B -flat-minor ,  and 
"'armonize  it  in  similar  manner. 


GOODRICH'S  ANALYTICAL  HARMONY.  133 

Chapter  XXXII. 


DIMINISHED  AND  CORRESPONDING  DOMINANT 
SEVENTH  CHORDS. 

THE  two  principal  discords  which  represent  a  given  key  are 
already  familiar.  One  is  founded  upon  the  leading-tone  of  a 
minor  scale ;  the  other  upon  the  5th  of  a  major  or  minor  scale. 
Three  notes  of  one  chord  occur  in  the  other.  The  root,  3d  and  5th  of 
the  diminished  yth  chord  are  the  same  as  the  3d,  5th,  and  yth  of  the 
dominant  yth  chord  to  the  same  key.  They  are  presented  together: 


Ex-  3IO-  rBtP*i|f  ^%t?r       Each  chord  appears  in  its  fundamental  posi- 


tion in  order  to  show  its  formation ;  but  it  would  not  be  proper  to 
employ  them  in  this  manner.  The  rules  of  progression  will  aid  in 
this  matter.  As  the  diminished  chord  is  to  come  first,  write  this  and 


tie  the  three  connecting  notes:     Ex-  3"-  F/rv-^^:^—       After  this  it 


is  apparent  that  the  yth  (/)  descends  a  minor  2d  to  <?.  This  will 
change  the  root  from  G-sharp  to  E,  as  another  kind  of  a  discord  re- 
sults. There  is  therefore  a  difference  of  but  one  tone  between  the 
two  chords.  Since  the  second  discord  belongs  to  the  same  tonic  as 
that  of  the  first,  the  author  names  the  second  chord  in  Ex.  310  the 
Corresponding  Dominant  yth.  The  natural  resolution  of  each  dis- 
cord will  show  the  propriety  of  this  nomenclature  : 


The  first  is  a  principal  diminished  yth  chord  resolved  to  its  tonic 
minor ;  at  (b)  the  Dominant  yth  chord  corresponds  to  the'diminished 
chord.  This  also  is  resolved  to  its  tonic  minor,  and  as  both  resolve 
to  the  same  minor  triad  they  are  corresponding  discords,  having 
three  tones  in  common. 

The  student  is  cautioned  against  confusing  the  corresponding 


134  GOODRICH'S  ANALYTICAL  HARMONY. 

dominant  yth  chord  with  the  antecedent  dominant  yth.     These,  with 
the  diminished  chord  in  the  center,  are  as  follows : 


Ex.  313. 


Antecedent  Douiiuaiit  7th.  Consequent  Dim.  7th.       Corresponding  Doni.  Till. 


The  antecedent  discord  belongs  to  C-major;  the  consequent  di- 
minished jth  to  A-minor;  the  corresponding  Dominant  yth  to  A- 
minor  or  A-major. 

The  student  is  to  write  a  diminished  yth  chord  on  the  leading- 
note  of  every  minor  scale  followed  by  its  corresponding  dominant 
7th  and  the  final  resolution  to  tonic  minor. 

The  next  would  be  to  E-minor.  The  leading  note  is  d-shaip, 
and  the  diminished  chord  is  to  contain  three  minor  3ds.  Retain  the 
root,  3d  and  5th,  move  the  jth  down  a  minor  2d  and  the  correspond- 
ing dominant  jth  chord  will  result.  This  is  resolved  to  tonic  minor 
and  the  example  is  complete.  See  illustration : 


The  figure  (i)  indicates  the  first  inversion  of  the  corresponding 
discord.  Observe  that  there  is  no  connecting  note  between  a  di- 
minished chord  and  its  tonic  resolution ;  whereas  the  root  of  the 
corresponding  discord  becomes  the  5th  of  the  tonic  triad.  This 
connecting  note  offers  an  advantage  in  final  resolutions  which  is  the 
principal  reason  for  using  the  corresponding  dominant  jth  chord. 

The  next  key  is  B-rninor.  The  root  of  the  diminished  chord 
will  be  a-sharp,  the  leading  tone  to  J3.  Build  the  chord,  change  it  to 
a  corresponding  dominant  jth,  and  resolve  naturally  to  tonic  minor. 

The  next  key  will  be  F-sharp-minor;  then  C-sharp-minor  and  so 
on,  till  all  the  minor  keys  have  been  represented.  Include  the  sig- 
nature to  each  key. 

By  writing  the  enharmonic  equivalent  of  D-sharp-minor,  the  stu- 
dent will  be  enabled  to  complete  the  cycle  of  minor  keys  by  fifths. 
E-flat,  B-flat,  F,  C,  G,  D,  A.  The  corresponding  dominant  jth  chord 
is  to  disappear  in  the  manner  already  explained,  this  being  the  sec- 
ond of  the  four  resolutions.  Direct  Cadence  and  Terminal  Resolu- 
tion are  therefore  synonymous.  Examples  for  comparison  may  be 
found  in  the  Key. 


GOODRICH'S  ANALYTICAL  HARMONY. 


135 


Chapter  XXXIII. 


THE    DIMINISHED   SEVENTH   CHORD    INVERTED. 
APPLICATION    OF    THE    CORRESPONDING 
DOMINANT    SEVENTH    CHORD.      IN- 
TERMEDIATE  AND  TERMINAL 
RESOLUTIONS. 

IN  order  to  show  the  results  of  inverting  a  diminished  yth  chord, 
and  the  necessity  for  introducing  a  corresponding  dominant  7th, 
the  different  inversions  will  be  given.     Suppose  we  begin  with  this 
chord : 


315. 


Its  natural  resolution  is  here  indicated,  and  as  the  root  of  the  con- 
cord is  in  the  base  it  may  be  considered  a  final  resolution. 

Write  the  3d  of  the  discord  as  a  real-base,  omitting  that  note 
from  the  upper  parts,  as  here : 


Ex.  316. 


Each  note  of  the  discord  is  to  be  resolved  according  to  previous 
instructions,  the  3d  alone  having  the  privilege  of  ascending  or  de- 
scending, according  to  circumstances.  Here  it  must  resolve  up  in 
the  ba-je  to  prevent  parallel  fifths.  The  root,  5th  and  jth  all  disap- 
pear according  to  formula.  The  result  to  be  particularly  noticed  is, 
that  the  concord  also  appears  inverted,  the  3d  being  in  the  base. 
And  as  the  final  concord  must  appear  uninverted  the  author  con- 


136 


GOODRICH  S    ANALYTICAL    HARMONY. 


siders  this  an  Intermediate  Resolution.  The  harmony  can  not  stop 
here,  but  must  continue  until  the  tonic  chord  appears  with  its  root 
in  the  base.  In  this  work  resolutions  are  classified  as  Intermediate 
and  Terminal.  The  last  example  must  therefore  be  continued. 
The  briefest  way  out  of  the  difficulty  is  this : 

Terminal. 


Ex.  317. 


The  first  chord  is  what  the  author  calls  the  Corresponding  Domi- 
nant yth  to  the  previous  diminished  yth  chord.  The  latter  belongs 
to  E-minor,  the  former  belongs  equally  to  E-minor  or  E-major.  The 
last  discord  resolves  direct  to  the  E-minor  triad  uninverted,  because 
no  5th  appears  above  the  real-base ;  therefore  its  resolution  is  not 


restrained,  the  5th         Ex-  3l8' 


having  become  a  4th 


Ex.  319. 


The  figures  in  the  base  refer  to  the  position  of 


each  chord  when  inverted.     The  last  discord  appears  in  its  second 
inversion.     In  Ex.  316  both  discord  and  concord  were  inverted. 

The  second  inversion  of  the  diminished  chord  is  now  selected  for 
resolution.     It  must  result  in  this  form  : 


Ex.  320. 


(1) 


Here  again  the  concord  appears  inverted,  for  the  5th  of  the  discord 
(a)  must  resolve  down  a  2d  to  the  3d  of  the  concord.  (See  figures  2 
and  i).  Accordingly  this  is  an  intermediate  resolution,  and  must  be 
followed  by  a  terminal  resolution.  As  it  is  not  well  to  skip  from  an 
inverted  base,  it  is  natural  to  choose  the  f-sharp  as  leading  melodic- 
ally  down  to  the  tonic.  The  final  cadence  will  thus  be  the  same  as 


GOODRICH'S  ANALYTICAL  HARMONY. 


in  Ex.  317.  (Perhaps  the  fact  should  again  be  noted  that  the  dupli- 
cated minor  3d  is  not  objectionable  when  it  results  from  the  resolu- 
tion of  two  different  parts  to  the  same  tone  of  the  concord,  both 
voices  moving  in  opposite  directions.  The  real-base  (a)  must  de- 
scend to  g,  and  the  f-sharp  above  also  resolves  to  g,  to  prevent 
parallel  fifths  with  the  c  above.) 

The  third  and  last  inversion  of  the  diminished  chord  comes  next. 
Omit  the  yth  from  the  upper  parts  and  proceed  to  resolve  each  note 
as  explained.  Cause  the  most  important  notes  of  the  discord  (i,  5, 
7)  to  disappear^  first,  because  their  resolution  is  prescribed.  *  *  * 
The  result  should  be  as  follows : 


Ex.  321. 


(3)     (2) 


The  f-sharp  might  have  gone  down  to  e,  as  no  5th  appears  above 
the  former  note.  The  resulting  triad  is  shown  to  be  in-  its  second 
inversion.  Certainly  it  can  not  rest  upon  this.  The  old  thorough- 
base  formula  may  be  employed  as  to  the  treatment  of  a  4  chord; 
i.  e.,  retain  the  real-base  as  root  of  the  dominating  harmony,  either 
with  or  without  the  yth.  Complete  this  terminal  cadence,  ending 
with  E  in  the  base.  *  *  * 

As  there  are  two  equally  proper  methods  of  accomplishing  this 
result  they  are  presented  for  reference : 


Ex.  322. 


As  a  final  ending  there  is  no  particular  choice  between  these.  (One 
of  these  is  supposed  to  be  joined  to  Ex.  321.) 

In  order  to  master  this  subject  it  will  be  necessary  for  the  stu- 
dent to  re-arrange  each  example  in  the  upper  parts,  without  altering 
the  real-base. 

An  outline  of  this  exercise  is  here  given,  to  be  filled  in  by  the 
pupil : 


GOODRICH  S    ANALYTICAL    HARMONY. 

^2L 


a 


Ex.  323. 


Wp  *    1?* 

r  —  M^— 

-     h-M 

\$Z- 

= 

A         *      ' 

"1 

^tt       H  —  i  — 

—  i  —  i  — 

r 

25=         J 

—  r  —  — 

—  — 

& 

J 

(1)                                                      (1)                     (U) 

4^ 

—  i 

,  15±£  

E  

-£  *  \~SS~ 

1   r  — 

1 



-*>--            1 

-^S>  1  \-^9 

E^, 

1 

S 

P  

\      \  1  \-±—\ 

-1  

F=l 

(•2)  (2)  (3)  (3)  (3) 

The  first  measure  shows  the  transition  chord  in  its  original  position. 
As  this  resolves  to  the  tonic  minor  triad  uninverted,  it  is  not  here 
necessary  to  work  this  out  by  means  of  the  corresponding  dominant 
7th.  It  is  included  to  show  the  chord  selected  for  illustration.  At 
(a)  the  first  inversion  is  to  be  resolved  to  tonic  minor  according  to 
directions.  This  will  be  an  intermediate  resolution,  and,  per  conse- 
quence, the  corresponding  dominant  jth  chord  is  to  be  introduced  in 
the  following  measure  and  resolved  as  a  terminal  cadence.  The 
next  example  (b)  is  the  same  inversion  re-arranged  above.  In  each 
instance  the  resolutions  are  similar,  with  exception  of  the  3d  of  the 
first  discord,  which  may  descend  a  whole  step  when  it  does  not  form 
a  5th  with  the  base  or  any  upper  part. 

The  root,  5th  and  yth  of  the  diminished  chord  have  fixed  resolu- 
tions, and  always  disappear  in  the  same  manner,  without  regard  to 
position  or  inversion.  This  fact  simplifies  the  task  considerably. 
Each  example,  as  (a),  (b),  or  (c),  is  to  be  complete  in  itself,  and  the 
last  chord  of  every  example  is  to  be  that  of  A-winor,  with  its  root  in 
the  base.  The  entire  modus  opcrandi  has  already  been  explained 
and  illustrated  with  the  diminished  chord  on  D-sharp.  Students 
will  therefore  have  no  trouble  in  completing  the  outlines  furnished 
by  Ex.  323.  *  * 

The  third  diminished  jth  chord  should  now  be  taken  for  similar 


illustration :     Ex-  324- 


The  inversions  will  consist  of  e, 


"The  present  intention  is  to  work  out  merely  the  three  primary  diminished  7th  chords, 
.  B,  and  C. 


GOODRICH'S  ANALYTICAL  HARMONY.  139 

g,  and  b-flat  in  regular  order  as  real-bases.  Each  inversion  is  to 
include  three  close  positions  in  the  upper  parts  as  explained.  The 
immediate  resolution  of  the  diminished  chord,  in  whatever  inversion, 
\vill  result  in  an  inversion  of  the  concord.  These  intermediate  reso- 
lutions must  be  followed  by  terminal  resolutions.  The  means  em- 
ployed are  the  same :  a  corresponding  dominant  7th  ending  with 
the  tonic  in  the  base. 

There  is  a  difference  of  but  one  note  between  the  two  principal 
discords  herein  employed : 


Ex.  325. 


Compare  the  two  discords  first,  and  then  their  respective  resolutions. 
In  the  second  measure  e  may  descend  to  d,  but  in  the  first  it  must 
ascend  to /I 

The  example,  when  completed,  ought  to  correspond  exactly  to 
the  other  two  illustrations,  A  and  B. 

The  student  should  finish  mis  task. 


Chapter  XXXIV. 


THEME  FOR  HARMONIZATION,  ILLUSTRATING  IN- 
TERMEDIATE AND  TERMINAL  RESOLUTIONS. 
FARTHER  VIEW  OF  INVERTED  BASES. 

A  CCORDIXG  to  the  previous  lesson,  all  resolutions  of  a  principal 
-*~^"  discord  are  intermediate  when  the  resulting  concord  appears 
inverted.  Therefore,  when  the  3d  or  5th  of  a  concord  occurs  as  real- 
base  we  must  proceed  until  a  terminal  resolution  is  effected.  The 
reason  for  this  is,  that  the  base  is  the  foundation  of  harmony,  and 
in  the  final  cadence  the  root  of  the  tonic  chord  must  appear  in  the 


140 


GOODRICH'S  ANALYTICAL  HARMONY. 


base  in  order  to  produce  a  sense  of  repose  or  completeness.  An; 
note  of  the  tonic  triad  may  appear  uppermost  and  leave  a  satisfac 
tory  impression,  provided  the  chord  reposes  upon  its  root.  Th 
diminished  chord  was  selected  as  illustrating  this  principle,  for  al 
its  inversions  resolve  to  an  inverted  concord.  But  the  dominant  71! 
chord  is  less  subject  to  this  influence ;  and,  as  a  matter  of  fact,  it 
root,  3d  and  5th  may  resolve  direct  to  the  tonic,  and  thus  constitut 
a  terminal  cadence.  An  example  of  this  is  appended : 


Ex.  326. 


-»- 


(1)      v         (2)       ^ 


Observe  particularly  that  the  tonic  appears  in  the  base  in  each  in 
stance,  and  yet  the  discord  is  resolved  strictly  according  to  formula 
The  only  inversion  that  results  in  an  intermediate  resolution  is  th 
third.  The  tendency  here  to  resolve  down  to  the  3d  of  the  concon 
is  too  strong  to  be  ignored : 


Ex.  327. 


(3)     (1)  (3)     (1) 

This  must  continue  until  a  more  complete  ending  is  reached,  as  here 


Ex.  328. 


(3)     (1)     (2      ^ 


Both  phrases  are  correct.  The  skip  of  a  4th  in  the  upper  part  at  (a 
is  desirable,  as  no  other  part  skips.  At  (b)  the  major  3d  is  doubled 
but  this  results  from  the  melodic  progression  of  the  theme  niovini 


GOODRICH'S  ANALYTICAL  HARMONY. 


141 


^^-^H 


contrarily  to  the  base.      Such   duplications  are   theoretically  and 
esthetically  correct. 

There  is  another  resolution  of  this  third  inversion,  in  which  the 
5th  of  the  discord  ascends  a  4th.  As  the  base  does  not  skip,  the 
soprano  may  assume  this  privilege  and  skip  from  5th  to  5th. 

Beethoven. 
J  I        ^ 


Jix.  329. 


(3)     (1)     (2) 

In  addition  to  the  connecting  note  the  3d  and  yth  of  the  discord  are 
resolved  most  naturally,  and  no  fault  occurs  as  a  result  of  this  unusual 
progression.  Besides,  the  half-open  position  of  the  B-flat  chord  is 
very  agreeable.  A  similar  instance  may  be  mentioned  in  connection 
with  the  following : 


Ex.  330. 


(2)     (1)     (2) 


Observe  the  final  resolution :  a  in  the  base  and  c  in  the  mezzo-so- 
prano part  descend  to  g  and  b-flat.  These,  being  tenths,  are  always 
euphonious;  the  leading  note  ascends  a  minor  2d,  thus  avoiding 
fifths  with  the  c  above.  The  soprano  part  skips  from  dominant  to 
tonic,  root  to  root.  In  this  instance  the  base  moves  alphabetically. 
Usually  this  is  more  effective  than  to  retain  the  5th  above  in  the  last 
chord.  (The  figures  indicate  the  inversions,  and  the  fermata  ^  shows 
where  the  cadence  is  complete.) 

The  last  four  examples  should  be  transposed  until  the  student  is 
familiar  with  these  applications.  *  *  * 

Before  presenting  the  illustrative  theme,  attention  will  be  called 
to  the  fact  that  the  corresponding  dominant  jth  may  be  introduced 
before  the  diminished  chord  is  resolved  to  its  tonic.  The  previous 
method  will  be  presented  first : 


GOODRICH  S    ANALYTICAL    HARMONY. 


A*     S      5      A 

_& 

a         &         1 

2-1 

VK 

! 

*/      F 
I       II 

1 

^~v  •           _j 

^        1 

B«      ^^ 

/p 

^g' 

c 

] 

1 

(2) 

Vi/ 

Ex.  331. 


The  diminished  chord  is  resolved  to  the  tonic  triad  in  the  seco 
measure,  the  5th  being  in  the  base.  This  necessitates  another  a 
more  final  resolution.  But  by  introducing  the  corresponding  ; 
chord  on  the  last  of  the  first  measure  it  may  resolve  directly  to  t 
tonic  chord,  and  so  end  : 


Ex.  332. 


ii  i 


(3)     (3) 

This  is  more  brief,  but  is  not  materially  different  from  the  exerci: 
in  Chapter  XXXII. 

The  yth  of  the  diminished  chord  resolving  down  a  minor  2d 
the  root  of  the  corresponding  dominant  yth  chord  may  occur  in  a 
of  the  parts,  though  the  resolution  will  not  always  be  terminal,  as 
the  last  example.  With  the  5th  of  the  diminished  chord  in  the  b; 
the  result  will  not  be  altered  by  introducing  the  corresponding  c 
cord,  because  that  interval  becomes  the  jth  of  the  second  chord, 
either  case  its  resolution  is  to  the  3d  of  the  concord  : 


Ex.333. 


The  result  is  an  intermediate  resolution  as  though  the  second  discc 
had  been  omitted.  With  the  3d  of  the  diminished  chord  the  res 
is  different : 


*The  Roman  numerals  indicate  the  kind  of  discord. 


GOODRICH'S  ANALYTICAL  HARMONY. 


Ex.  334. 


(1)     (1) 


At  (a)  the  concord  appears  inverted,  and  is,  accordingly,  intermediate. 
But  by  introducing  the  corresponding  discord  at  (b)  the  resolution 
is  directly  to  tonic.  The  explanation  is,  that  the  5th  between  the 
base  and  soprano  in  the  first  discord  is  changed  to  a  4th  in  the  second 
discord.  These  results  will  be  the  same  in  the  various  positions  of 
this  inversion.  These  should  be  written  by  the  student.  *  *  * 
A  theme  will  now  be  presented  in  which  the  newly  acquired 
information  is  to  be  applied : 


Ex.  335. 


0    (1-)    (2J 


(3)    (3) 


In  s'-ipplying  the  middle  parts  of  the  harmonization  one  should  be 
governed  principally  by  the  base.  The  figures  apply  only  to  inverted 
chords,  b  being  numbered  (2)  signifies  that  the  root  is  a  5th  below. 
Therefore  the  yth  is  in  the  melody,  and  as  this  is  to  be  a  dominant 
7th  chord  the  remainder  is  easily  supplied.  The  Roman  numerals 
indicate  the  antecedent,  diminished,  or  corresponding  dominant  jth 
chords  in  this  order  : 

I  refers  to  an  antecedent,  followed  by  a  consequent  discord  when- 

ever the  numerals  I,  II  follow  each  other  successively. 

II  always  refers  to  the  diminished  chord. 

I  following  II  indicates  a  corresponding  dominant  7th. 


GOODRICH'S  ANALYTICAL  HARMONY. 

In  moving  the  base  to  or  from  an  inversion  it  is  advisabL 
preserve  as  much  as  possible  a  melodic  progression  in  that  p 
This  is  equivalent  to  the  remark  that  the  base  must  not  skip  exc 
from  root  to  root.  In  the  present  exercises  this  is  unexception 
true,  and  as  these  independent  melodic  progressions  in  the  base  ] 
are  generally  attractive  and  afford  variety  to  the  fundamental  ba 
it  would  be  well  to  cultivate  them,  especially  since  their  proper  n 
agement  is  much  more  difficult  than  the  original  base  movement  fi 
root  to  root.  But  this  is  intended  neither  as  a  rule  nor  a  prohibit 
It  is  simply  a  direction  for  the  student's  present  guidance.  The  ol 
arrangements  should  be  made  from  the  last  example,  without  altei 
the  base  part.  Transpose  into  various  keys. 

The  design,  though  occupying  only  a  short  period,  is  com] 
hensive,  and  the  author  considers  it  worthy  of  thorough  tre<itm< 
The  solution  may  be  found  in  the  Key. 


b    ANALYTICAL,    HAK,>IONY. 


PART  VIII. 


Chapter  XXXV. 


PRINCIPAL  AND  SECONDARY  SEVENTH  DIS- 
CORDS.   THEIR  ORIGIN,  APPLICATION, 
AND  EFFECT. 


THE  dominant  and  diminished  yth  chords  are  to  be  classed  as 
Principal  discords.  That  is,  they  contain  the  elements  of  tran- 
sition, and  when  resolved  naturally  are  capable  of  performing  a  de- 
cided terminal  cadence.  Nearly  all  yth  chords  in  a  given  key  are 
secondary.  These  lack  the  modulatory  elements  and  are  incapable 
of  effecting  a  modulatiot  or  of  performing  a  cadence. 

A  yth  chord  is  built  upon  every  degree  of  the  normal  scale : 


Ex.  336. 


Here  are  five  species  of  discord.  I  and  II  we  know  to  be  transition 
chords.  The  remainder  will  be  classified  in  this  manner :  The  dis- 
cord founded  upon  the  tonic  contains  a  major  3d,  normal  5th  and 
major  yth.  This  is  the  harshest  of  all  the  yth  chords.  Number  it  Y. 
Is  there  another  discord  in  this  scale  of  the  same  species?  \Yhat 
is  it?  This  should  also  be  numbered  V.  The  discord  founded  upon 
the  second  of  the  scale  contains  a  minor  3d,  normal  5th  and  minor 
yth.  This  is  less  harsh,  and  is  numbered  IV.  Are  there  other  clis- 


t46  GOODRICH'S    ANALYTICAL    HARMONS 

cords  in  the  example  containing  the  same  intervals?  If  so,  they 
belong  to  the  same  species,  and  are  to  be  numbered  identically.  The 
chord  upon  the  7th  of  the  major  scale  is  still  less  discordant.  It  is 
numbered  III.  It  consists  of  a  minor  3d,  imperfect  5th  and  minor 
7th,  and  may  be  used  as  a  secondary  or  as  a  principal  discord. 

The  diminished  chord  comes  next.  This  is  an  agreeable  discord, 
and  is  numbered  II,  as  in  a  former  chapter.  It  can  be  used  in  a 
major  key,  but  it  occurs  naturally  in  the  harmonic  minor  scale. 
Hence  it  is  included  among  the  unaltered  discords. 

The  most  agreeable  of  all  discords  (so-called)  is  the  one  founded 
upon  the  5th  of  the  scale,  the  familiar  dominant  yth  chord.  This 
was  the  first  discord  explained,  and  being  the  most  important  it  is 
numbered  I.  Corresponding  Roman  numerals  should  be  written 
above  each  chord  in  Ex.  336. 

The  discords  marked  V  are  so  harsh,  if  sounded  independently, 
that  they  must  be  submitted  to  some  process  of  preparation.  As 
this  is  explained  elsewhere,  the  author  will  merely  give  directions 
concerning  the  present  application  of  these  dissonances.  The  dis- 


sonant interval  is  the  major  yth  :     Ex<  337-    fry    ^  ~        and,  as  with- 


out  the  yth  there  results  a  consonant  major  triad,  it  may  be  concluded 
that  the  b  is  the  disturbing  element.  In  order  to  prepare  the  ear 
for  this  dissonance,  begin  with  a  consonant  chord  containing  b,  and 
gradually  introduce  the  c  either  above  or  below  : 

IL 


Ex.  338. 


The  base  begins  with  a  consonant  root-note  and  moves  down  by- 
natural  degrees.  The  first  step  results  in  a  moderate  discord,  duly 
prepared.  The  second  step  (d  to  c),  though  perfectly  natural,  results 
in  a  discord  of  the  fifth  species.  But  the  suspended  notes  above, 
together  with  the  objective  progression  of  the  base  part,  have  a 
tendency  to  ameliorate  the  dissonating  effect  of  this  harsh  combina- 
tion. And  what  is  more  important,  the  base,  progressing  by  scale- 
^like  degrees,  while  the  other  parts  remain  passive,  presents  a  distinct 
object  in  what  transpires,  and  is  its  own  raison  d'etre. 


GOODRICH  S    ANALYTICAL    HARMONY. 


The  dissonant  interval  may  be  introduced  above  by  causing  the 
upper  voice-part  to  descend  a  half-step,  while  the  lower  parts,  as 
consonances,  remain  stationary : 


Ex.  339. 


The  first  chord  is  purely  consonant,  and  while  the  three  lower  parts 
are  retained,  the  soprano  merely  descends  a  minor  2d.  Any  disso- 
nance, however  harsh,  would  be  justifiable  if  produced  in  this  manner, 
on  account  of  the  seemingly  slight  difference  between  the  consonant 
and  dissonant  harmonies.  This  difference,  though  theoretically  very 
great,  is  merely  what  is  heard  or  perceived  in  this  example, 


Ex.  340.  j^    p?     ^__         while  the  same  fundamental  harmony  pre- 

*) 

vails.     These  principles  will  apply  to  all  harsh  combinations,  though 
the  author  doubts  the  propriety  of  preparing  discords  in  general. 

Before  proceeding,  attention  is  directed  to  the  fact  that  in  both 
the  previous  examples  three  notes  remain  passive  while  one  moves 
alphabetically.  This  plan  will  be  followed  in  the  practical  applica- 
tion of  these  discords. 

Each  of  the  secondary  yth  chords  (including  the  secondary  tran- 
sition chord  on  the  yth  of  the  scale)  should  be  inverted,  as  students 
must  be  familiar  with  all  positions  of  these  discords.  Omit  the  base 
note  from  the  upper  parts.  One  of  these  is  presented  as  a  sample  : 


Ex.  341. 


:Eg= 


(1)         (2)       (3) 

The  V  sho  A-S  the  species  and  the  figures  indicate  the  inversion. 
Follow  the  same  plan  with  the  other  inverted  discords. 


143  GOODRICH'S  ANALYTICAL  HARMONY. 

A  brief  illustration  will  be  given  of  the  manner  in  which  these 
secondary  discords  are  to  be  utilized  : 


—     *~-,Z-  i  55-  -55 

^       xj 

1 

fa  " 

—  (2  £      h-  p 

.  —  £ 

w^ 

—  j  i  H  1— 
V         IV 

_L_  ,  ,  . 
^n  i      i 

^Mtyt              ,y                   f 

i         (* 

+ 

3      & 

te= 

ff  f  '  

^  r- 

EE 

1  

H 

(3) 

(1)     W 

Ex.  342. 


At  V  a  harsh  discord  results  in  the  manner  already  explained.  From 
here  the  degree  of  dissonance  is  gradually  reduced  by  moving  only 
one  part  at  a  time  until  the  dominant  yth  is  reached.  This  is  an 
euphonious  discord.  Number  II  is  omitted,  being  unnecessary  here. 
In  progressing  from  one  concord  to  another,  or  in  resolving  the  essen- 
tial yth  chord,  one  connecting  note  is  sufficient. 

Re-arrange  this  in  the  upper  parts  and  transpose,  as  it  is  prelimi- 
nary to  what  follows. 

In  harmonizing  the  following  theme  three  notes  are  to  be  tied  in 
each  succeeding  chord  of  the  dissonant  progressions.  The  moving 
notes  may  occur  in  any  voice-part,  above  or  below.  Real-bases  are 
not  to  be  duplicated  here,  and  in  the  dissonant  chords  even  the  root- 
tone  must  not  be  doubled.  (See  inversions,  Ex.  341.) 


Ex.  343. 


IV 


IV         V        IV         III        II 


(3)       o      (1)     (2) 


(1) 


Previous  directions  as  to  the  movement  of  voice-parts,  together 
with  the  accompanying  numerals,  will  enable  the  student  to  complete 
this  in  correct  manner,  especially  since  the  number  of  each  inversion 
is  indicated  by  figures  below.  Roots  in  the  base  are  marked  with 
ciphers. 

The  entire  series  of  yth  chords  in  the  chart,  Ex.  336,  is  to  be 
employed  in  this  harmonization,  and  the  theme  is  so  designed  that 
the  dissonances  are  gradually  reduced  in  harshness  after  the  second 
discord  marked  V.  Observe  the  numerals.  *  *  * 


GOODRICH'S  ANALYTICAL  HARMONY. 


149 


The  next  step  consists  in  writing  the  contralto  part  uppermost  as 
theme.  Use  the  same  chords  and  the  same  base. 

Another  resulting  theme  may  be  obtained  by  copying  the  original 
mezzo-soprano  part,  and  writing  the  harmony  beneath  it.  These  two 
themes  are  given : 

Ex.  344. 


ffi  

^=5: 


This  exercise  is  susceptible  of  still  another  arrangement,  and  as  it 
illustrates  a  useful  principle  it  will  be  included  here.  Reference  is 
made  to  the  original  base  part  (from  third  measure)  being  transferred 
to  the  soprano  as  theme : 

Ex.  345. 


V        IV        III      II 


The  first  two  measures  of  the  original  theme  are  not  adapted  to  the 
base,  but  from  the  third  measure,  where  the  parts  are  bound  so 
closely  together,  any  of  the  upper  parts  might  appear  below.  The 
only  difference  in  treatment  would  be  required  in  the  final  cadence. 
In  case  the  last  note  was  a  real-base  (inversion  of  the  tonic  chord) 
it  would  be  necessary  to  make  some  alteration.  For  instance,  in  the 
theme  marked  (c)  the  last  chord  could  not  have  its  5th  below,  but 
would  proceed  from  root  to  root,  thus : 


Ex.  346. 


(3)     (8 
This  theory,  as  well  as  the  resolution  of  the  diminished  chord  into 


150 


GOODRICH'S  ANALYTICAL  HARMONY. 


the  corresponding  dominant  7th,  has  been  set  forth  in  a  previous 
chapter. 

Transpose  these  exercises  into  various  other  major  scales. 

*     *     # 

These  secondary  and  principal  yth  chords  may  also  be  treated 
fundamentally,  especially  if  they  form  a  harmonic  sequence.  In 
sucl»  situations  the  secondary  discords  assume  the  functions  of  a 
principal  yth  chord,  and  are  apparently  resolved  in  the  same  man- 
ner ;  but  it  must  be  understood  that  the  former  merely  occupy  cer- 
tain positions  in  the  scale  according  to  the  nature  of  the  sequence, 
and  that  their  resemblance  (on  paper)  to  the  principal  discords  is 
apparent,  not  real. 

In  rococo  music  this  treatment  was  of  common  occurrence  on 
account  of  the  infrequency  of  transition  passages.  Since  the  advent 
of  Boccherini  and  Mozart  the  tendency  has  been  to  change  the 
secondary  into  principal  yth  chords.  An  example  of  the  former 
method  will  be  quoted  as  of  greater  present  consequence: 


It  is  from  a  Rondo  by  Paganini.     The  discord  marked  V  is  resolved 
(disappears)  as  though  it  appeared  as  a  principal  discord  on  c-flat : 


Ex.  348. 


But  in  the  original  the  a  and  d  are  natural, 


and  the  chords  are  all  confined  to  the  scale  of  B-flat  until  the  tempo- 
rary modulation  to  the  dominant  in  the  last  measure.  The  chord 
marked  V  is  very  harsh,  but  here  it  is  duly  prepared  by  the  previous 
b-flat  and  d,  and  it  also  results  naturally  as  part  of  the  sequence :  f, 
e-flat,  d,  c,  in  the  base. 

The  discord  marked  IV  may  be  explained  in  like  manner,  though 
the  base  here  ascends  from  d  to  g,  the  same  as  does  the  root  of  a 
dominant  yth  chord.  In  the  resolution  of  the  discord  on  e-flat  the 
root  ascends  an  augmented  4th,  because  the  normal  4th  (a-flat)  is 


GOODRICH  S    ANALYTICAL    1IAKMONV.  151 

not  a  part  of  the  B-flat  scale.     Observe  that  the  5th  of  the  triads  are 
omitted.     If  these  be  added  the  progressions  would  appear  like  this: 


Ex.  349 


These  more  closely  resemble  the  original  treatment  of  these  discords 
as  connecting  links  and  suspensions. 

The  student  should  particularly  note  the  design  of  the  sequence 
in  Ex.  347 ;  how  the  discords  take  their  places  in  the  sequence  upon 
successive  degrees  of  the  natural  scale ;  how  the  secondary  chords 
disappear  as  they  follow  the  model,  and  the  effect  of  the  entire  pas- 
sage. This  will  prove  much  more  useful  than  all  the  rules  of  thor- 
ough-baser books,  for  it  discloses  the  motive,  and  forms  the  basis  for 
all  available  musical  knowledge.* 

Transpose  Ex.  347  into  several  major  scales.  If  this  can  be  done 
at  the  piano  prima  vista,  it  will  render  unnecessary  the  act  of  writing 
the  notes. 


Chapter  XXXVI. 


ADDITIONAL  CHORD  PROGRESSIONS. 


previous  lessons  in  chord  progression  have  demonstrated 
-A-  the  fact  that  the  movement  of  a  concord  can  not  be  prescribed. 
Certain  principles  can  be  deduced  from  standard  compositions  and 
according  to  logical  theory  ;  but  it  can  not  be  said  that  a  certain 
tone  of  a  certain  concord  must  be  followed  by  a  certain  other  tone 
of  another  chord. 

I'.c-in  with  the  concords  containing  a  given  tone.     These  are  six 
in  number. 


Ex.  350.  bfc= 


*See  accompaniment  to  Schumann's  "  Ich  grollc  nicht. 


152 


GOODRICH'S  ANALYTICAL  HARMONY. 


The  first  three  occur  in  A-major,  the  last  three  in  A-minor.  But 
they  are  all  related  in  a  secondary  manner  to  the  key  of  A.  Trans- 
pose into  G  and  B-flat,  and  add  the  fundamental  bases.  The  direc- 
tions for  harmonic  progression  would  apply  to  any  of  these  chord 
successions. 

Observe  the  following  miscellaneous  triad  progressions : 

Ex   351. 


vTZ     ^ 

=zg^=|= 

l^S 

-2,^=L 

:_^"-:^J 

~-%3£\ 

:^T2?- 

\-4^-*~ 

*'                                                                                                       '  (5*™ 

a                  b                 c                    d                   e                  f                  g 

<Y    H  —  i 

1  

H  —  «-i 

-—  ^  's 

—1  -r- 

—  1  (2  — 

^—d— 

25*  

i  »4  —  - 

^-^q 

—  »  

—0—\  

?E&§ 

Here  are  twenty-one  consonant  chord  progressions  beginning  with 
the  triad  of  C-major.  They  have  all  been  employed  in  actual  com- 
position, and  all  are  conducted  according  to  previous  directions.  It 
would  be  a  useful  practice  for  the  student  to  re-arrange  this  example 
in  two  other  positions,  and  transpose. 

The  general  principles  governing  the  movement  of  chords  are  of 
almost  universal  application.  Though  these  doctrines  and  motives 
have  been  somewhat  thoroughly  explained,  the  author  has  deemed 
it  wise  to  present  no  deviations  from  a  reasonable  formula,  unless 
such  deviation  permitted  a  simple  explanation  of  its  exceptional  char- 
acter. But  the  student  is  supposed  to  be  familiar  with  the  motives 
governing  harmonic  progressions,  and  may  now  consider  certain 
harmonic  movements  purposely  omitted  in  the  previous  chapters. 
One  of  these  is  in  reference  to  consecutive  octaves,  such  as  these : 


GOODRICH  S   ANALYTICAL    HARMONY. 


In  the  first  measure  the  lower  parts  move  from  g  to  a  ;  but  as  there 
is  no  harmony  between  these  parts  the  result  is  a  mere  re-enforce- 
ment of  the  base  or  of  the  lower  treble  part.  In  the  next  measure 
the  soprano  part  is  doubled  above,  these  parts  being  independent  of 
the  others.  These  progressions  are  correct.  But  if  inverted,  they 
will  be  unsatisfactory : 


Ex.  353. 


This  is  not  good,  and  it  will  be  necessary  to  employ  one  of  the 
following  methods,  as  already  advised  in  the  chapter  on  Avoided 
Cadences  : 


Ex.  354- 


&•• 


'2,        ST 


At  (a)  a  simple  progression  in  contrary  movement  occurs.  At  (b) 
the  soprano  and  base  move  up  in  tenths,  while  the  two  middle  parts 
descend  in  thirds.  Such  movements  are  always  good.  Resolutions 
that  result  in  a  half-open  position  as  at  (b)  may  continue  in  this  form, 
provided  they  are  correct  with  the  base : 


Ex.  355. 


No  capable  critic  will  object  to  such  arrangements.    Here  is  another 
of  similar  character,  with  the  movements  reversed  : 


GOODRICH'S  ANALYTICAL  HARMONY. 


Ex.  356. 


These  are  of  a  contrapuntal  nature,  since  the  design  is  not  so  muc! 
a  series  of  regular  chords  in  progression  as  it  is  to  preserve  th 
sequence  of  thirds  and  sixths  above,  while  the  base  moves  up  wit) 
a  melodic  progression  against  the  descending  upper  parts.  At  <  a 
there  results  a  mere  duophonic-chord,  e  and  g;  that  is,  it  does  no 
naturally  suggest  or  represent  a  certain  tri-chord,  as  does  the  due 
phonic-chord  at  (c).  In  writing  a  full  score  w  litre  more  harmon; 
was  desired  c,  b-flat,  or  c-sharp  might  be  added  to  the  two-fold  chon 
on  e.  Any  of  these  \vould  be  correct : 


Ex.  357. 


1 

J. 


In  the  second  measure  there  are  parallel  fifths,  yet  several  justifiable 
circumstances  favor  them  :  i.  The  fullness  of  the  harmony;  2,  tin 
fifths  being  in  the  middle  parts ;  3,  the  contrary  and  decided  progres 
sion  of  the  base  part. 

To  return  to  Ex.  356.  At  (b)  a  seventh  chord  of  the  third  specie 
is  resolved  as  a  principal  discord,  the  3d  being  omitted.  This  chon 
will  appear  again  in  this  chapter.  At  (c)  the  sequence  comes  to  ai 
end,  because  the  same  design  could  not  be  carried  beyond  this  poin 
without  seeming  to  be  forced.  In  fact,  the  sequence  would  ordinaril; 
end  on  the  G  chord  between  (b)  and  (c). 

With  regard  to  the  secondary  transition  chord  founded  on  the  jtl 
of  a  major  scale,  there  are  three  methods  of  employing  it : 

1.  As  a  secondary  discord  in  connection  with  suspensions.     Ii 
this  sense  it  is  not  resolved  directly,  but  two  or  more  of  its  tone 
remain  passive  as  parts  of  another  discord.     Such  was  the  applica 
.tion  in  Chapter  XXXV. 

2.  It  may  be  resolved  in  an  intermediate  manner,  thus : 


GOODRICH  S    ANALYTICAL    HARMONY. 

ib-  r— 


155 


Ex.  358. 


These  may  be  classed  as  progressions.  The  first  example  is  fre- 
quently used  in  a  minor  cadence.  Its  resolution  is  so  apparent,  and 
depends  upon  so  many  circumstances,  that  it  need  not  be  described. 
The  example  at  (b)  has  been  employed  by  Rubinstein  in  Ballet  Music 
to  "  Feramors,"  and  explains  itself.  A  few  other  instances  of  these 
two  intermediate  resolutions  follow : 


— r  r    LA P3=q 

jg=[=!=pbz=pi^^ 


The  only  explanation  necessary  is  with  regard  to  the  second  arrange- 
ment at  (a) :  «,  c,  e  become  g,  b,  e.  This  is  simple  enough.  But 
the  base  skips  up  a  fourth  instead  of  resolving  to  g,  thus  assuming 
the  position  of  a  dominant,  skipping  from  the  .root  of  the  discord  to 
the  real-base  on  b.  Or  it  might  go  from  root  to  root : 


Ex.  360. 


c  and  e  descend  to  b  and  d-sharp,  a  remains  as  yth  of  the  principal 
discord,  while  the  base  moves  in  a  contrary  direction. 

3.     The  discord  under  notice  may  resolve  directly  to  the  chord 
of  which  its  root  is  the  leading  tone : 


Ex.  361. 


m     » 

1 

\-\)       t 

1         i 

V                    I 

1 

+                +       +         4 

1         1         ' 

+      +       + 

m        m       * 

~       0 

i          i 

-^                   i         i 

156  GOODRICH'S   ANALYTICAL    HARMONY. 

When  used  in  this  manner  as  a  principal  discord,  the  root  should  be 
in  the  base,  and  not  be  doubled  above.  The  three  arrangements 
here  exhibited,  together  with  the  one  in  Ex.  345,  are  all  that  the 
luthor  recommends.  Transpose  these.  The  effect  of  a  direct,  ter- 
minal resolution  of  this  discord  is  rather  mild  and  sunny.  It  has 
less  character  and  decision  than  the  direct  resolution  of  discords  ] 
or  II. 

By  selecting  the  tonic  or  dominant  as  a  stationary  tone  two  othei 
parts  may  move  in  thirds  or  sixths  in  any  direction  to  or  from  tht 
stationary  tone : 


Ex.  362. 


This  is  so  elementary  and  so  pleasing  as  to  require  no  harmonic 
analysis.  It  is  a  progression  in  thirds  against  a  stationary  tone  thai 
forms  a  part  of  the  dominant  or  tonic  chords.  These  are  marked  witl: 
a  -f .  A  fourth  part  may  be  added  by  doubling  the  base  or  the  sta 
tionary  tone.  Or  the  dominant  yth  chord  may  be  used  throughout 
considering  the  e  and  g  as  passing  tones  between  the  intervals  of  the 
discord,  thus : 


Ex.  363. 


It  is  not  desirable  to  change  the  harmony  on  the  second  and  fourtl 
beats  in  such  instances.  Observe  that  the  two  lower  parts  move  ir 
tenths  while  the  root  and  yth  remain  above.  The  contralto  melodj 
would  be  equally  appropriate  in  the  soprano  part : 


Ex.  364. 


GOODRICH'S  ANALYTICAL  HARMONY.  757 

The  tonic  may  be  selected  as  a  stationary  note : 


Ex.  365. 


The  same  principle  is  observable  here  with  regard  to  the  parts  moving 
in  tenths.  Each  third  (or  tenth)  is  necessarily  a  part  of  some  chord, 
though  such  passages  are  not  treated  the  same  as  full  chord  progres- 
sions. The  stationary  note  might  be  an  octavo  lower,  or  it  might 
be  placed  as  a  pedal  note  in  the  real-base  part,  thus : 


Ex.  366. 


The  lower  G  is  related  to  all  the  chords  above.  The  chords  marked 
-f ,  though  entirely  satisfactory  here,  are  not  theoretically  complete. 
The  student  might  attempt  to  supply  the  remaining  note  in  these 
chords,  but  without  employing  the  same  design.  This  harmonic 
representation  is  important,  therefore  an  explanatory  example  will 
be  included  in  the  Key.  *  *  * 

In  passing  from  the  tonic  concord  to  any  of  the  related  dominant 
yth  chords,  no  difficulty  will  be  experienced  as  most  of  these  have 
already  been  employed.  They  are  presented  in  the  most  usual  forms : 


Ex.  367. 


After  the  discord  is  introduced  it  may  be  resolved  directly,  indirectly 
or  intermediately. 

We  now  pass  to  the  three  diminished  chords  belonging  naturally 
to  this  key : 


158  GOODRICH'S  ANALYTICAL  HARMUXY. 

Ex.  368. 


The  third  discord  (fifth  and  sixth  measures)  belongs  to  D-minor 
By  changing  the  c-sharp  to  d-flat  it  may  go  to  F-major,  as  in  the  last 
measure.  The  treatment  of  these  diminished  chords  has  been  ex 
plained.  These  subjects  will  be  farther  illustrated  in  chapters  or 
Pedal-Note  and  Harmonic  Counterpoint. 

In  conclusion  it  is  advised  that  all  these  examples  be  transposed 


Chapter  XXXVII. 


SUCCESSION  OF  DOMINANT  SEVENTH  CHORDS 
DIATONICALLY  AND  CHROMATICALLY. 

ANOTHER  MEANS  OF  TRANSITION. 

THE  fact  was  demonstrated  in  a  previous  chapter  that  discord^ 
do  not  always  resolve  to  concords. 

A  succession  of  related  dominant  yth  chords  is  given  in  whict 
there  are  two  connecting  notes : 


The  3d  and  5th  of  the  first  chord  remain  passive  and  become  5th  and 
yth  of  the  second  discord.  The  process  of  this  change  is  simple: 
The  root  of  the  first  discord  is  sharpened,  while  the  yth  descends  a 
minor  2d.  The  second  discord  may  resolve  to  E-major  as  well  as 
to  E-minor,  presenting  a  simple  method  of  performing  more  distant 
transitions.  By  selecting  the  principal  yth  chord  on  the  super-tonic 
we  may  accomplish  a  similar  result : 


GOODRICH  S    ANALYTICAL    HARMONY. 


15? 

r-  -fi-  5-  -&.—*-&    r-  '*& . 

Ex.  370. 

The  final  resolutions  are  naturally  minor ;  the  major  cadence  merely 
shows  the  possibilities  of  these  resultant  discords.  These  illustra- 
tions conclude  with  a  similar  change  of  the  discord  of  the  first  species 
on  the  tonic : 


This  results  in  the  same  manner  and  produces  a  similar  effect. 

This  process  is  to  be  carried  out  with  the  other  dominant  jth 
chords,  founded  upon  3,  6  and  7  of  the  major  scale.  Simply  raise 
the  root  a  chromatic  step,  lower  the  7th  a  minor  2d  and  resolve  to 
tonic,  major  or  minor.  The  following  example,  containing  dominant 
7th  chords  founded  upon  every  degree  of  the  E-flat  scale,  is  to  be 
worked  out  in  the  same  manner.  Some  of  the  final  resolutions 
should  be  major  and  some  minor : 


Ex. 


All  these  admit  of  re-arrangement  and  inversion,  which  should  be 
included  in  the  student's  examples.  The  original  3d  and  5th  remain 
stationary  in  every  instance,  and  become  5th  and  7th  of  the  second 
discord.  *  :;:  * 

\Ve  are  now  at  liberty  to  move  from  one  dominant  7th  chord  to 
any  other,  provided  there  is  a  connecting  note.  This  license  includes 
all  the  essential  discords  except  two :  those  situated  a  minor  second 
above  and  below  any  given  dominant  7th  chord : 


These  the  author  would  exclude,  for  they  involve  consecutive  fifths 
and  consecutive  minor  sevenths,  which  are  still  more  objectionable. 
The  5th  might  be  omitted  or  placed  in  the  base,  but  the  incongruity 
would  still  be  manifest.  Any  of  the  others  may  be  employed  in  suc- 
cession, and  in  certain  situations  all  might  be  made  effective.  The 
connecting  note  or  notes  must  remain  as  such,  and  objectionable 
movements  are  to  be  avoided.  Here  follow  some  of  the  more  un- 
usual 7th  chord  successions : 


ibo 


GOODRICH'S  ANALYTICAL  HARMONY. 


Ex.  374- 


(a)  and  (b)  of  No.  r  are  practicable.  The  second  $ih  at  (c)  is  imper 
feet,  and  so  it  is  admissible,  though  inferior  to  (a)  and  (b).  Nuin 
ber  2  is  still  better,  and  may  be  used  freely  in  any  position  or  inver 
sion.  Number  3,  though  more  remote,  is  equally  effective.  Then 
are  two  connecting  notes  (including  b,  c-flat),  and  the  5th,  g  and  a 
moves  contrarily  to  a-flat  and  d-flat.  The  same  appears  enharmon 
ically  at  (c).  The  fourth  example  is  also  good,  and  may  appear  it 
other  positions.  No.  5  is  somewhat  forced,  and  would  be  more  effect 
ive  in  a  pedal  passage  than  as  a  succession  of  fundamental  discords 
The  other  position,  with  the  /uppermost,  is  not  recommended. 

These  miscellaneous  chord  successions  are  presented  with  a  viev 
to  independent  transitions,  usually  of  an  abrupt  nature.  For  instance 
if  this  passage  were  introduced  : 


Ex.  375. 


the  next  strain  could  be  in  A-flat  (major  or  minor), 
be  said  of  examples  i,  2,  3  and  5. 


The  same 


NATURAL  SUCCESSION  OF  SEVENTH  CHORDS. 

The  most  usual  progression  of  essential  discords  is  that  in  whicl 
the  root  of  each  chord  resolves  up  a  4th  or  down  a  5th.  Referena 
is  made  to  a  sequence  of  yth  chords,  not  to  a  progression  to  some  par 
ticular  one  for  a  transitional  purpose.  For  instance,  a  discord  of  thi 
first  species  naturally  resolves  to  the  concord  a  4th  above  (domina:; 
to  tonic),  as  here  : 


Ex.  376. 


GOODRICH'S  ANALYTICAL  HARMONY. 


161 


This  is  the  first  resolution  of  an  essential  discord,  in  which  it  pur- 
sues its  natural  tendency.  If  to  the  second  chord  is  added  a  minor 
7th  there  will  result  two  discords  of  the  same  species  following  in 
regular  order,  thus : 


Ex.  377. 


The  only  difference  between  this  and  the  previous  example  is  that  the 
3d  of  the  first  discord  descends  a  chromatic  step  in  place  of  ascending 
a  minor  2d.  Compare  the  examples.  Another  yth  chord  may  follow 
in  the  same  manner;  i.  e.,  all  the  notes  of  the  discord  are  resolved 
according  to  formula  excepting  the  3d,  which  descends  a  chromatic 
step.  Write  this.  *  *  * 

The  same  description  will  apply  to  each  progression,  and  the  base 
will  ascend  a  4th  or  descend  a  5th  during  the  continuance  of  the  se- 
quence. The  fundamental  base  to  these  successive  7th  chords  has 
been  frequently  utilized  by  composers  as  a  base  solo,  and  such  pro- 
gressions always  attract  attention  by  their  independence  of,  and  dis- 
similarity to,  the  other  parts.  If  we  employ  but  four  parts,  retaining 
the  fundamental  progression  below,  it  will  be  necessary  to  omit  the 
root  and  the  5th  alternately.  When  the  root-note  is  doubled  above, 
it  will  remain  as  5th  of  the  next  discord.  In  this  case  the  connecting 
note  will  appear  alternately,  thus : 


Ex.  378. 


m 


No  exception  to  the  previous  resolution  here  occurs.  If  we  begin 
writh  a  discord  in  its  complete  form  the  result  will  be  the  same,  ex- 
cepting that  the  connecting  note  will  be  between  the  second  any 
third  chords : 


Ex.  379. 


162 


GOODRICH'S  ANALYTICAL  HARMONY. 


As  the  root-note  does  not  appear  above  in  the  first  chord  there  is  no 
connection.  But  the  chords  follow  in  such  natural  sequence  that 
the  connecting  note  is  not  indispensable.  Of  the  last  two  arrange- 
ments (b)  is  perhaps  more  satisfactory. 

The  student  may  supply  the  harmony  to  the  following  motive 
considered  as  a  base  solo,  using  only  dominant  yth  chords  until  the 
final  cadence  : 


Bass  Sequence  as  Theme. 


Ex.  380. 


(2) 


All  the  bases  marked  I  represent  uninverted  dominant  yth  chords. 
The  ciphers  indicate  consonant  roots.  (2)  refers  to  the  second  inver- 
sion of  the  tonic  chord.  This  is  used  in  order  to  make  the  cadence 
perfect.  Necessary  chromatic  signs  are  of  course  to  be  included.  It 
is  not  of  material  consequence  what  position  the  upper  harmony 
assumes,  provided  it  is  not  too  high  nor  too  low.  The  sequence 
continues  through  three  measures,  but  the  sixth  chord  is  to  be  con- 
sonant. *  *  * 

Write  two  arrangements  of  this,  one  beginning  as  at  (a),  the  other 
as  at  (b)  :     *    *     * 


Ex.  381. 


By  omitting  the  fundamental  sequence  in  the  base  and  substituting 
alternate  real-bases,  we  can  preserve  a  connecting  note  throughout : 


Ex.  382. 


lizfczd 


(2) 


(2) 


The  first  two  chords  are  connected  by  the  base  ;  the  second  and  third 
by  the  contralto.  This  is  accomplished  in  the  same  way  ;  /.  c.,  the 
root-note  remains  passive  and  becomes  the  5th. 

The  figures  in  the  base  show  that  every  other  chord  appears  with 
its  5th  below.  This  arrangement  is  more  smooth,  but  less  character- 
istic than  where  the  base  proceeds  fundamentally.  The  base  part  ta 


GOODRICH'S    ANALYTICAL    HARMONY. 


163 


the  last  exercise  may  be  given  the  soprano,  the  original  contralto 
part  appearing  in  the  base,  thus : 


Ex.  383. 


(2) 


(2)        o 


This  is  simply  a  re-arrangement  of  the  previous  example.  The  whole 
notes  above  are  of  course  equivalent  to  connecting  notes.  Transpose 
the  last  two  illustrations  into  D,  E-flat  and  F.  *  *  * 

These  successive  7th  chords  will  now  be  considered  as  a  means 
of  harmonizing  a  descending  chromatic  melody : 


Ex.  384. 


^ 


This  is  to  be  harmonized  in  the  manner  already  explained.  The 
beginning  and  ending  are  not  so  easily  managed  as  the  succession 
of  discords.  Therefore  these  parts  have  been  indicated.  It  would 
be  well  to  write  one  arrangement  with  fundamental  bases  and  another 
in  which  the  base  moves  alphabetically,  every  other  chord  being  an 
inversion.  In  following  the  latter  plan  the  student  may  find  it  con- 
venient to  know  that  when  the  root  of  the  discord  is  omitted  above, 
that  tone  belongs  in  the  base ;  and  when  the  5th  is  omitted  above,  it 
is  to  be  included  as  a  real-base.  Transpose  the  theme  and  harmo- 
nize accordingly. 

In  conclusion,  the  author  would  suggest  that  these  successive 
dominant  yth  chords  should  not  continue  indefinitely,  on  account  of 
their  transitional  nature.  Even  one  of  them  alone  suggests  resolu- 
tion, and  when  several  succeed  one  another  the  ear  is  more  or  less 
deceived  by  their  indirect  resolution,  and  unconsciously  longs  for  a 
more  satisfying  cadence.  Some  appreciable  motive  must  therefore 
appear  as  a  justification  for  continuing  these  essential  discords  beyond 
the  limits  of  the  previous  examples.  They  constitute  a  species  of 
avoided  cadence. 


164  GOODRICH'S  ANALYTICAL 


PART  IX. 


Chapter  XXXVIII. 


FIFTEEN  ENHARMONIC  TRANSITIONS  BY  MEAN' 

OF  THREE  PRIMARY  DIMINISHED  SEVENTH 

CHORDS.    THEORY  OF  NOTATION. 

A  S  a  diminished  yth  chord  may  be  founded  upon  the  leading-tom 
-^~*-  of  every  minor  scale,  and  as  there  are  but  three  diminished  ytl 
chords  that  are  essentially  different,  it  is  evident  that  each  diminishe< 
chord  can  be  made  to  represent  four  different  keys,  in  addition  to  thi 
enharmonic  equivalents.  These  three  primary  discords  and  thei 
natural  resolutions  are  familiar. 

That  all  other  chords  of  this  species  are  inversions  or  enharmonl 
representations  of  these  three  original  chords  may  be  seen  from  thi 
illustration : 

Ex.  385. ' 


Every  chromatic  tone  in  the  octave  is  embraced  by  these  thre< 
chords.  The  next  chord  above  the  last  would  result  in  an  inver 
sion  of  the  first  chord. 

The  chord  of  the  diminished  7th  is  a  chromatic  harmony  com 
posed  of  three  minor  3ds.  Select  the  chord  (A)  and  proceed  with  it: 
resolution.  When  the  chord  is  based  upon  thirds,  as  in  the  first  posi 
tion,  the  root  is  to  be  considered  as  leading-tone,  and  the  chord  natu 
rally  belongs  to  the  minor  key  situated  a  minor  2d  above  the  root 
Each  note  in  the  diminished  chord  nny  be  considered  as  a  leading 
note.  Therefore  the  four  minor  keys  which  this  cnord  may  represen 
are  situated  a  minor  2d  above  each  of  these  notes : 


GOODRICH'S  ANALYTICAL  HARMONY. 


165 


Ex.  386.  hi£z_- 


I? 


What  are  these  four  tonics  ?     The  resolution  of  the  first  is  known  to 
be  this : 


Ex.  387. 


Now  consider  B  as  a  root  and  build  upon  it  three  minor  thirds.  If 
correctly  formed  the  upper  note  (yth)  will  appear  as  the  enharmonic 
-equivalent  of  g-sharp.  Resolve  this  to  tonic  minor  according  to  pre- 
vious directions  and  number  it  2.  D  becomes  the  next  root.  Build 
three  minor  thirds  upon  this  note  and  do  not  confuse  the  augmented 
2d  with  the  minor  3d.  This  is  to  be  resolved  to  tonic  minor  and 
numbered  3.  The  necessary  chromatic  alterations  are  to  be  supplied 
according  to  the  signature  of  the  key  to  which  the  discord  resolves. 
No.  4  begins  upon  Fas  root.  Add  three  minor  thirds  above  this  root- 
note,  and  resolve  to  tonic  minor  as  usual.  The  enharmonic  equiva- 
lent of  No.  4  should  then  be  written  in  sharps ;  /"appearing  as  e-sharp. 
This  is  No.  5.  It  is  remarkable  that  this  single  discord  may,  by  a 
mere  enharmonic  change  in  the  appearance  of  its  intervals,  be  made 
to  represent,  equally  five  different  minor  keys.  The  last  transition 
will  illustrate  this  fact  : 


Ex.  388. 


The  discord  and  its  resolution  comprise  every  note  in  the  scale  of 
G-flat  minor,  as  may  be  seen  by  comparing  the  two.  The  scales  of 
A-tninor,  C-minor,  and  E-flat  minor  yield  the  same  results.  *  *  * 
The  diminished  chord  (B)  is  selected  and  submitted  to  the  same 
process.  Each  of  these  notes  become  roots : 


Ex.  389.  : 


and  every  time  a  new  root  is  selected  the  upper  note  is  to  be  enhar- 
monically  changed  from  its  former  appearance.  The  tonics  are  to  be 
£,  G,  B-flat,  D-flat  and  C-sharp.  If  the  discord  is  noted  correctly  it 


1 66 


GOODRICH  S    ANALYTICAL    HARMONY. 


will  contain  a  minor  3d,  imperfect  5th,  and  diminished  yth  from  th 
root,  and  will  belong  naturally  to  the  minor  scale  a  minor  2d  abov 
the  root,  which  is  the  leading-tone.  The  harmonic  form  of  the  mino 
scale  is  here  presupposed.  The  student  should  complete  this  task  ii 
the  manner  indicated  for  chord  (A).  *  *  * 

The  diminished  jih  chord  (C)  comes  next.  It  must  be  made  t 
represent  equally  the  keys  B-minor,  D-minor,  F-minor,  A-flat  mino 
and  G-sharp  minor.  These  will  comprise  the  entire  fifteen  mino 
keys.  No.  4  in  each  example  should  be  written  enharmonically  a 
a  simplification.  For  instance,  No.  4  of  chord  (A)  requires  for  it 
signature  nine  flats.  For  the  convenience  of  the  performer,  con 
posers  usually  write  this  in  three  sharps : 


Ex.  390. 


Compare  this  with  Ex.  388.  No.  5  (B)  is  to  be  written  in  four  sharp 
in  place  of  eight  flats,  and  number  5  (C)  is  to  be  written  in  five  sharp 
in  place  of  seven  flats : 


Ex.  391. 


In  each  instance  5  is  a  simplification  of  4 ;  otherwise  they  are  synonj 
mous.  During  the  five  enharmonic  changes  of  each  primary  dimir 
ished  yth  chord  the  sounds  remain  the  same.  The  metamorphosis  i 
therefore  only  in  appearance.  A-flat  is  substituted  for  g-sharp ;  i 
natural  becomes  c -flat ;  A-sharp  becomes  b-flat,  and  so  on,  according 
to  the  notation  required  by  the  scale  of  the  key  to  be  established 
To  make  this  still  plainer  the  enharmonic  changes  of  chord  (A)  ar 
presented  in  such  manner  that  each  chord  may  be  sounded  by  th 
same  keys  of  a  piano  or  organ  : 


Ex.  392-  bfc= 


The  root  of  i  is  G-sharp;  the  root  of  2  is  B ;  of  3,  D ;  of  4,  / 
Therefore  by  selecting  any  key  of  which  one  of  these  tones  may  b« 
come  the  leading-tone,  and  writing  the  discord  according  to  the  hai 
momc  minor  scale  of  that  key,  it  might  resolve  directly  to  any  o 
these  minor  chords  as  tonic  harmonies :  A,  C,  E-flat  or  G-flat.  Prac 


GOODRICH'S  ANALYTICAL  HARMONY.  167 

tically  the  four  chords  in  the  last  example  are  the  same ;  theoretically 
they  are  different,  and  represent  four  different  keys. 

A  practical  illustration  of  this  may  be  seen  in  the  principal  move- 
ment of  Beethoven's  Op.  13.  It  occurs  in  the  brief  episode,  Grave, 
that  precedes  the  development.  A  diminished  7th  chord  on  f-sharp 
is  first  resolved  to  G-minor ;  and  in  the  next  measure  the  e-flat  is 
changed  to  d-sharp  and  leads  to  E-minor,  the  key  in  which  the  devel- 
opment begins. 

The  next  step  is  to  resolve  each  discord  in  the  last  example  with 
fr-sharp  or  a-flat  lowermost.  The  original  exercises  in  which  each 
chord  appeared  in  its  first  position  may  be  consulted.  The  first 
arrangement  is  given  as  a  model : 

"-*-1— *~^— -^fefcdto^qfe 


cz-      — — 


T 

Each  discord  is  resolved  according  to  formula.  The  root,  5th  and 
7th  have,  as  heretofore,  a  fixed  resolution ;  the  3d  usually  ascends. 
But  when  it  is  above  the  7th  the  3d  may  descend.  This  resolution 
is  given  whenever  it  does  not  involve  parallel  fifths.  (See  third 
measure,  last  example.) 

In  the  second  arrangement  the  lowest  note  will  be  b  or  c-flat. 
Write  these  and  resolve  regularly.  The  third  arrangement  has  d,  or 
e-doublc-jlat  lowermost  during  the  four  (or  five)  measures.  In  the 
fourth  and  last  arrangement  f  or  e-sharp  will  appear  throughout  the 
example  as  real-base.  Resolve  each  discord  according  to  the  actual 
requirements,  and  include  the  enharmonic  equivalent  of  No.  4,  as 
shown  in  Kx.  393. 

The  other  two  discords,  (B)  and  (C),  should  undergo  the  same 
process,  for  it  is  important  that  students  should  be  thoroughly  famil- 
iar with  the  relations  and  possibilities  of  this  remarkable  chord  in  all 
its  phases.  Besides,  this  is  the  surest  way  of  mastering  the  theory 
of  notation. 

Any  note  of  the  diminished  chord  may  appear  in  the  base  and 
resolve  according  to  the  directions  in  Chapter  33. 

See  solutions  to  Chapters  33  and  38  in  the  Key. 


i68 


GOODRICH'S  ANALYTICAL  HA.RMONY. 


Chapter  XXXIX. 


THE  DIMINISHED  SEVENTH  CHORD  AS  A  MEANS 
OF  ENHARMONIC  TRANSITION.     CHRO- 
MATIC HARMONIZATION. 

THE  student's  field  of  operations  has  been  so  greatly  enlarged 
that  hereafter  it  will  not  be  necessary  to  establish  any  other 
boundary  than  that  of  Propriety. 

The  diminished  yth  chord  is  susceptible  to  a  greater  variety  of 
uses  than  any  other  discord.  With  a  single  diminished  chord  we 
may  pass  directly  to  four  minor  or  four  major  keys.  The  theory  of 
this  metamorphosis  has  been  explained.  The  practical  application 
will  now  be  sought.  In  this  connection  it  will  frequently  be  neces- 
sary to  employ  the  corresponding  dominant  7th  as  a  means  of  decid- 
ing the  tonality ;  for  while  a  discord  of  the  second  species  may  repre- 
sent with  equal  elements  four  minor  keys,  a  discord  of  the  first  species 
is  naturally  associated  with  but  one  tonic.  Heretofore  the  corre- 
sponding dominant  yth  was  used  as  a  matter  of  convenience  in  resolv- 
ing to  tonic  minor  from  an  inverted  diminished  chord.  This  privi- 
lege is  still  available,  though  the  primary  object  of  changing  the 
species  of  discord  from  II  to  I  will  now  be  to  determine  and  estab- 
lish the  tonality  wherever  that  is  doubtful.  Owing  to  the  chromatic 
character  of  the  diminished  chord  it  is  liable  (especially  if  two  or  more 
of  these  chords  be  used  in  succession)  to  unsettle  our  mental  concep- 
tion of  the  actual  key-tone ;  or,  what  is  more  serious,  it  might  create 
an  impression  directly  contrary  to  that  intended  by  the  composer. 
Examine  the  following : 


Ex.  394. 


The  tonality  is  more  or  less  doubtful  after  the  third  chord,  which  can 
be  made  to  represent  by  its  notation  four  different  tonics.    As  it  here 


GOODRICH'S  ANALYTICAL  HARMONY.  169 

appears  the  last  chord  represents  D -minor ;  by  changing  the  b-flat  to 
a-sharp  it  would  belong  to  B ;  if  the  c-sharp  appeared  as  d-flat  it 
would  resolve  to  F-minor  ;  finally,  if  the  c-sharp  and  e-natural  were 
noted  as  d-flat  and  f-flat,  it  would  represent  A-flat.  But  it  must  be 
remembered  that  these  proposed  changes  in  notation  merely  alter  the 
appearance  of  the  chord ;  they  do  not  in  any  way  affect  the  sounds 
practically.  The  very  possibilities  of  this  chord  present  themselves 
as  an  obstacle  here ;  for  since  it  may  resolve  with  equal  propriety  to 
four  tonics  it  can  not  belong  particularly  to  any  of  these  keys.  Here 
is  where  the  corresponding  dominant  jth  chord  serves  the  highest 
purpose.  It  decides  the  tonality  and  anticipates  or  locates  the  tonic 
with  certainty,  even  before  the  discord  disappears. 

The  process  to  be  undergone  in  making  this  change  from  a  dis- 
cord of  the  second  to  the  first  species  has  been  explained.  The  dis- 
cord II  must  first  be  so  noted  as  to  suggest  by  its  appearance  the  key 
to  .be  established.  Then  merely  lower  the  yth  a  minor  2d  and  the 
tonality  is  established. 

The  solution  follows : 


Ex.  395. 


g^TTfeg^FT^S-VH 


The  tonics  to  these  may  be  major  or  minor.  If  the  original  tonality 
still  lingers  in  the  mind,  resolve  number  3  naturally  to  F-minor,  and 
No.  4  to  A-flat  major — these  being  related  keys  to  C-minor.  Acting 
upon  this  hypothesis  the  first  measure  would  resolve  to  D-minor 
(being  nearer  to  C-minor).  It  would  be  difficult  to  decide  the  mode 
of  number  2  theoretically ;  for  the  minor  3d  (d)  belonged  to  the  orig- 
inal key,  and  the  major  3d  (d-sharp}  might  be  considered  as  the 
enharmonic  equivalent  of  e-flat. 

But  in  reality  the  key-impression  is  so  far  erased  by  the  third 
chord  that  either  mode  may  be  chosen  in  any  of  these  instances. 
The  main  point  to  determine  is  the  key-tone ;  the  mode  is  of  second- 
ary importance.  A  similar  example  is  presented,  the  different  solu- 
tions of  which  are  to  be  worked  out  by  the  student : 


170  GOODRICH  S    ANALYTICAL    HARMONY. 


Ex.  396. 


The  last  chord  is  to  be  changed  in  its  enharmonic  appearance  so  as 
to  represent  A,  C,  E-ftat  and  F-sharp.  Then,  before  the  resolutions, 
the  corresponding  discords  are  to  be  introduced  and  resolved  to  their 
respective  tonics,  major  or  minor. 

When  the  corresponding  yth  chord  appears  in  its  3d  inversion 
the  resolution  will  be  intermediate,  and  accordingly  the  terminal 
resolution  is  to  be  introduced  afterwards.  This  will  serve  to  recall 
a  previous  lesson. 

The  intermediate  resolution  may  sometimes  be  minor  and  the 
final  cadence  major.  One  example  of  this  will  be  given: 


Ex.  397. 


This  is  a  continuation  of  Ex.  396.  When  a-flat  descends  to  g  the 
tonality  is  naturally  decided  as  that  of  C.  After  the  inverted  minor 
chord  the  base  passes  to  d  on  its  course  down  to  the  final  tonic. 
Many  instances  like  this  occur  in  which  the  terminal  resolution  is 
major,  affording  another  proof  of  the  remark  previously  made  that 
the  key-tone  is  of  much  more  consequence  than  the  mode.* 

The  corresponding  jth  chord  resolving  to  C  is  the  only  one  of 
the  four  that  results  in  an  intermediate  resolution.  In  the  other 
three  the  base  will  proceed  at  once  to  the  tonic. 

CHROMATIC  HARMONIZATION. 

The  dominant  yth  chord  has  already  been  utilized  in  the  harnio 
nization  of  a  descending  chromatic  theme.  The  diminished  chord 

'•'  Until  theTniddle  of  the  iMh  century  it  was  customary  after  writing  a  composition  in  a 
minor  key  to  make  the  final  cadence  in  tonic  major.  The  object  was  not  to  perform  a  tran- 
sition, but  to  observe  a  rule  then  in  vogue  that  the  minor  chord,  being  an  imperfect  conson- 
ance, was  therefore  unsuitable  for  the  repose  of  a  final  cadence. 


GOODRICH'S  ANALYTICAL  HARMONY. 


171 


can  be  used  for  chromatic  themes  both  descending  and  ascending. 
And  this  suggests  a  few  remarks  concerning  the  chromatic  scale,  and 
its  notation.  This  is  an  incidental  scale,  not  naturally  associated 
with  any  particular  key-tone.  It  may  ascend  or  descend  in  rapid 
succession  without  destroying  a  recognized  tonality,  especially  if  a 
fundamental  harmony  be  employed  as  accompaniment.  In  these 
instances  it  is  customary  to  use  sharps  ascending  and  fiats  descend- 
ing, thus : 


The  notation  here  is  merely  a  matter  of  convenience  or  simplicity, 
the  object  being  to  employ  as  few  chromatic  signs  as  possible.  But 
where  the  chromatic  tones  are  harmonized  separately  the  pupil  must 
be  governed  by  the  natural  notation  of  the  chords,  and  the  prevailing 
tonality.  An  example  will  illustrate  this  : 


.Vet  in. 


Ex.  399. 


oto. 


The  chromatic  motive  is  here  accompanied  by  five  essential  discords, 
each  of  which  contains  a  major  3d,  normal  5th,  and  minor  yth  from 
its  root.  Therefore  the  notation  must  correspond  to  these  harmonic 
relations  and  can  not  be  a  mere  matter  of  convenience.  Only  one 
flat  sign  is  employed,  and  this  b~flat  is  sub-dominant  to  the  key  of  F, 
to  which  the  discord  resolves.  Every  chromatic-  tone  here  affects 
the  tonality,  because  the  chromatic  tones  are  not  treated  as  incident- 
al, passing  tones,  but  each  one  is  harmonized  with  a  transition  chor.d 
containing  the  altered  note.  In  writing  a  succession  of  diminished 
chords  the  notation  is  more  a  matter  of  convenience ;  for  the  chord 
itself  is  a  chromatic  one,  and  as  it  does  not  ordinarily  represent  a  cer- 
tain key-tone  (as  does  the  essential  discord)  its  representation  is  not 
prescribed  by  its  fundamental  note.  Therefore  (a)  is  better  than  (b)  : 


172 


GOODRICH  S   ANALYTICAL    HARMONY. 


Ex.  400. 


because  fewer  chromatic  signs  are  employed.  But  when  the  dimin- 
ished chord  is  used  as  a  fundamental  harmony  its  notation  should  be 
governed  by  the  scale  to  which  it  belongs : 


Ex.  401. 


The  first  discord  belongs  to  tonic  minor ;  the  second  one  goes  to  D- 
minor.  Each  diminished  chord  is  noted  accordingly.  These  direc- 
tions and  suggestions  with  regard  to  the  theory  of  notation  will  cover 
all  the  ground  to  be  traversed  at  present. 

The  harmonization  of  a  chromatic  theme  by  means  of  diminished 
chords  is  effected  by  employing  an  entirely  different  chord  to  accom- 
pany each  melodic  tone.  This  would  seem  to  be  a  strange  procedure, 
since  all  the  parts  move  up  or  down  the  same  distapce  and  without 
a  connecting  tone.  One  of  these  chromatic  progressions  will  be  an- 
alyzed. The  chord  is  composed  of  a  minor  3d,  imperfect  5th,  and 
diminished  jth.  The  minor  thirds  may  follow  one  another  with  good 
effect,  as  this  example  proves : 


Ex.  402. 


The  next  interval  is  an  imperfect  5th.  This  is  the  same  in  sound  as 
an  augmented  4th.  There  is  no  objection  to  a  succession  of  these 
intervals,  especially  if  they  are  accompanied  by  minor  thirds  above 
or  below,  or  by  a  holding  tone,  thus  : 

Ex.  403. 

» 

This  necessarily  admits  a  progression  of  two  minor  thirds,  as  here : 

Ex.  404. 


I          I    -1 — [- — ffp— — 1 


GOODRICH'S  ANALYTICAL  HARMONY. 


In  these  instances  the  augmented  2d  is  synonymous  with  the  minor 
3d.  The  interval  of  a  diminished  jih  comes  next.  As  this  is  the 
same  in  sound  as  a  major  6tk  there  can  be  no  objection  to  a  succes- 
sion of  these  : 


Ex.  405. 


On  account  of  the  notation  the  first  and  third  intervals  are  diminished 
sevenths,  the  second  and  fourth  being  major  6ths.  But  as  they  began 
with  a  diminished  7th,  and  as  each  part  ascends  chromatically,  the 
effect  is  the  same  as  if  the  f  -sharp  and  g-  sharp  had  been  noted  as 
g-flat  and  a-flat.  Certain  of  the  discords  appear  inverted  as  they  suc- 
ceed each  other,  and  this  is  why  the  intervals  of  a  diminished  7th  do 
not  follow  one  another  theoretically.  Another  noticeable  feature  of 
these  chromatic  chords  is  that  they  may  follow  one  another  in  any 
position  and  in  any  direction.  The  best  effects  are  produced  by 
omitting  the  base  tones  from  the  upper  chords,  and  this  plan  will  be 
followed.  Take  a  descending  chromatic  theme  : 


(1)   o 


This  begins  and  ends  in  C-minor,  and  the  tone  omitted  in  the  second 
chord  is  to  be  included  in  the  base.  From  this  point  all  the  parts 
descend  chromatically.  In  the  fifth  measure  the  diminished  7th  chord 
must  be  so  represented  that  its  root  will  be  the  leading  note  to  C. 
In  the  last  of  this  measure  the  corresponding  dominant  7th  is  to  be 
introduced  by  lowering  the  diminished  7th  a  minor  2d.  The  (i)  be- 
low signifies  the  first  inversion  of  the  essential  discord.  The  posi- 
tions selected  are  to  be  one  of  these  : 


Ex.  407. 


LIZ?- 


After  harmonizing  the  theme,  transpose  it  to  B-flat  minor  and  arrange 
in  the  same  manner.     *     *     * 

A  chromatic  theme  ascending  is  now  to  be  harmonized.     The 
same  theory  here  applies  to  the  other  parts: 


174 


GOODRICH'S  ANALYTICAL  HARMONY. 


Ex.  408. 


The  succession  of  diminished  chords  begins  upon  the  second  quarter 
of  the  first  measure  and  continues  to  the  last  of  the  third  measure. 
The  first  chord  in  the  last  measure  must  necessarily  be  a  correspond- 
ing dominant  yth,  not  alone  to  decide  the  tonality,  but  to  make  a 
terminal  cadence. 

Complete  the  harmonization  and  transpose  to  A  and  D. 

In  a  former  chapter  a  succession  of  essential  yth  chords  was  em- 
ployed as  a  means  of  harmonizing  descending  chromatic  themes. 
Ascending  chromatics  were  not  harmonized  in  this  manner  because 
it  would  be  unnatural  and  directly  contrary  to  the  progressive  ten- 
dency of  the  essential  discord.  The  author  has  observed  a  few  iso- 
lated cases  of  this  kind,  but  they  were  very  brief,  or  of  a  forced  char- 
acter. There  is  such  an  instance  in  the  Prayer  and  Barcarolle  from 
Meyerbeer's  "  Star  of  the  North"  : 
Ex.  409.  , 

s*^^.  li.w~~   s—  .     ^-_ 


lE*-~- 

0t 

H    ,• 

i  —  j* 

i^5±       Eg 

-g  tf 

p-5  —  '*- 

!   P\   111- 

W  — 

51T* 

0      i     |                                        0                         -             '                    - 

^:         * 

ft 

-^  • 

f 

(J 

jf^  * 

B 

*£  —  ^  

_j  

r-H- 



r        ? 

There  are  two  circumstances  that  tend  to  make  this  tolerable  :  The 
two  discords  are  separated  by  the  half  rest  in  the  second  measure, 
and  only  two  discords  of  the  first  species  are  employed.  As  a  means 
of  harmonizing  two  or  more  chromatic  tones  the  order  will  be  directly 
reversed  in  all  ascending  passages,  and  the  author  does  not  advise 
young  composers  to  imitate  this  last  example. 

There  is  another  mode  of  treating  the  diminished  chords  when 
they  accompany  a  chromatic  theme.  The  upper  parts  may  proceed 
as  before  by  half  steps  while  the  base  moves  in  an  opposite  direction 
by  whole  steps.  •  This  will  not  prevent  the  latter  from  forming  a  part 
of  each  chord  above  it.  A  brief  example  of  this  will  suffice : 


Ex.  410. 


GOODRICH  S    ANALYTICAL    HARMONY. 


175 


All  but  two  of  the  real-bases  are  doubled  above ;  but  the  diminished 
chords  are  not  here  treated  fundamentally,  and  the  base  moves  con- 
trarily  to  the  upper  parts. 

Chromatic  passages  have  frequently  been  harmonized  by  means 
of  inverted  major  chords.  Such  an  instance  is  the  following  from 
Chopin's  E-minor  Concerto : 


Ex.  411. 


*     T 


This  is  a  sequence  of  major  chords  in  the  second  position,  the  real- 
base  being  omitted  from  the  design.  The  stamp  of  Chopin  is  suffi- 
cient to  establish  this,  or  any  of  his  chord  successions,  as  right  and 
proper,  but  these  major  chords  in  chromatic  progression  are  so  bold 
and  incisive  that  they  should  be  used  carefully  and  sparingly.*  In 
the  last  example  they  occur  in  such  rapid  succession  that  the  tonality 
is  not  interfered  with. 


Chapter  XL. 


THE  DIMINISHED  SEVENTH  CHORD  AS  A  PASSING 

HARMONY  TO  THE  TONIC  AND  TO  THE 

DOMINANT  SEVENTH. 


TO  THE  TONIC. 


TN  previous  examples  the  diminished  jth  chord  has  been  resolved 
•*-  directly  to  tonic  minor,  to  another  diminished  chord,  or  changed 
to  a  corresponding  yth  chord  of  the  first  species.    The  latter  is  a  par- 
tial progression  ;  the  former  is  a  principal  resolution.    In  the  follow 
ing  example  the  diminished  chord  performs  satisfactory  cadences  : 


GOODRICH'S  ANALYTICAL  HARMONY. 


Ex.  412 


The  elements  of  transition  here  are  b,  f,  and  a-flat.  These  being  re- 
solved naturally  constitute  a  regular  resolution  or  cadence.  But  if 
we  select  a  diminished  7th  containing  c,  no  elements  of  transition  to 
C  will  appear : 


Ex.  413- 


The  diminished  chord  is  here  treated  as  a  passing  harmony,  and  in 
this  secondary  resolution  the  tonality  of  c  is  not  affected.  Therefore 
if  these  melodic  notes  occur : 


Ex.  414. 


I_*0 fL_ 


tlie  d-sharp  and  f-sharp  may  be  considered  as  parts  of  the  passing 
diminished  harmony  without  any  transitional  tendency.  The  reso- 
lution of  these  passing  chromatic  tones  is  indicated  in  the  exercise, 
and  as  c  belongs  to  both  chords  it  may  be  retained  as  a  note  of  con- 
nection : 


Ex.4i5. 


The  chord  of  C-major  is  considered  as  tonic,  c  being  the  principal 
tone  throughout.  The  chromatic  passing  tones  are  thus  accom- 
panied without  disturbing  the  key-impression,  and  in  this  sense  the 
diminished  chord  is  used  as  a  passing-harmony — its  resolution  being 
secondary. 

Other  positions  of  the  passing  diminished  chord  on  the  tonic  are 
here  given : 


Bx.  416. 


GOODRICH'S  ANALYTICAL  HARMONY. 


177 


The  progression  is  designated  as  a  passing  diminished  chord  on  the 
ionic.  Tonic  here  refers  to  the  unaltered  major  key-tone,  not  to  the 
tonic  of  the  diminished  chord,  which  would  be  E  if  the  discord  were 
resolved  as  a  transition  chord.  A  few  simple  directions  for  this  sec- 
ondary resolution  of  a  diminished  chord  are  given. 

The  first  figure  represents  a  tone  of  the  discord,  and  the  figure 
following  the  dash  shows  the  resolution  to  some  part  of  the  concord, 
thus :  i  —  i  (this  represents  the  tonic,  which  remains  stationary). 
2  sharp  —  354  sharp  —  5  ;  6  —  5.  This  chart  will  apply  to  any  ma- 
jor key.  Therefore  the  following  melodic  notes  can  be  accompanied 
with  the  passing  diminished  chord  of  wrhich  any  of  these  notes  form 
a  part :  1,2  sharp,  4  sharp,  6.  This  presupposes  that  2  sharp  goes 
to  3,  4  sharp  to  5,  or  6  to  5.  The  tonic  is  usually  included  above, 
as  it  may  be  doubled  freely.  The  following  arrangements  are  most 
practicable,  and  should  be  transposed  into  several  keys : 


Ex.  417. 


6  —  5 


As  two  notes  of  the  discord  resolve  to  the  5th  of  the  concord,  there 
is  usually  a  choice  of  arrangement  in  harmonizing  the  former.  Either 
of  the  last  two  measures,  for  instance,  may  be  used.  By  omitting 
the  tonic  above,  there  results  a  half-open  position  of  the  tonic  triad. 
This  would  be  equally  effective  in  such  instances  as  these : 


Ex.  418. 


In  the  second  measure  4  sharp  goes  down  to  3,  and  not  up  to  5,  but 
no  objection  to  this  could  be  sustained.  The  other  arrangements 
(Ex.  417)  are  more  conveniently  handled. 

The  following  theme,  when  harmonized  as  intended,  will  illustrate 
the  manner  in  which  a  diminished  chord  is  employed  in  a  secondary 
capacity,  as  a  passing  harmony : 


GOODRICH'S  ANALYTICAL  HARMONY. 


Ex.  419. 


The  secondary  resolutions  are  indicated.  Every  tone  of  the  discord 
is  introduced  in  the  theme,  the  resolutions  being  according  to  the 
previous  formula.  The  second  and  seventh  measures  are  to  contain 
the  dominant  yth  chord,  though  in  the  cadence  this  should  be  pre- 
ceded by  the  sub-dominant  harmony. 

Transpose  into  A-flat. 

Any  note  of  the  discord  may  appear  in  the  base,  though  this  will 
result  in  an  inversion  of  the  tonic  chord  in  all  arrangements  other  than 
the  i  —  i  heretofore  employed.  Resolve  the  base  as  though  its  note 
appeared  above.  This  requires  much  care  in  the  management  of  the 
chords  preceding  and  following  the  diminished  harmony.  The  ex- 
amples presented  will  serve  as  models  until  the  student  has  become 
independent  of  professional  assistance : 

Ex.  420. 


As  an  intermediate  progression  the  arrangement  at  (a)  is  simple  and 
effective.  The  next  two  are  employed  mostly  in  final  cadences.  At 
(c)  the  diminished  chord  is  preceded  by  a  secondary  yth  chord  having 
two  notes  in  common.  Write  two  more  arrangements  without  alter- 
ing the  base,  and  transpose  to  A  and  B. 

PASSING    DIMINISHED   CHORD   TO  THE   DOMINANT    SEVENTH. 

A  diminished  chord  containing  the  root  of  a  dominant  yth  chord 
may  be  used  according  to  the  same  theory  here  applied  to  the  tonic 
chord  and  its  passing  diminished  harmony.  In  both  instances  the 
note  of  connection  is  the  root  of  the  resultant  harmony — tonic  or 
dominant  7th.  Begin  with  the  fundamental  position  of  the  essential 
discord : 


Ex.  421. 


Arrange  in  two  other  close  positions.     * 


GOODRICH  S    ANALYTICAL    HARMONY. 


179 


In  any  key  or  position  the  notes  of  a  diminished  chord  are  situ- 
ated a  minor  2d  below  the  3d,  5th  and  yth  of  the  essential  discord, 
the  dominant  note  connecting  the  two  discords.  It  is  easy  to  deduce 
from  this  a  formula  that  will  apply  to  any  key,  thus :  i  sharp  —  2, 
3  —  4,  5  —  5,  6  sharp  7,  computing  from  the  prevailing  tonic.  If 
these  notes  occur  melodically : 


Ex.  422. 


add  the  remaining  intervals  according  to  the  formula,  thus : 


Ex.  423. 


This  also  is  a  secondary  resolution  of  the  diminished  chord.  The 
dominant  should  be  duplicated  above  when  the  principal  discord  is 
to  be  resolved  directly.  But  if  an  avoided  cadence  is  contemplated 
the  root-note  of  the  essential  discord  is  not  to  be  included  above. 
These  two  methods  are  here  illustrated : 


Ex.  424. 


=^^3=*$ 


etc. 


-zr 


In  the  avoided  cadence  at  (a)  the  root-note  is  omitted  above.  At  (b) 
the  root-note  is  in  the  soprano  part  as  the  principal  discord  is  re- 
solved directly.  The  position  of  the  passing  diminished  chord  must 
be  governed  by  the  position  of  the  principal  discord.  In  the  second 
measure  the  base  has  6  sharp,  which  resolves  to  7.  These  inversions 
are  good  when  they  result  from  a  melodic  progression.  When  they 
result  from  a  skip  they  are  usually  less  effective.  The  following  ex- 
ercise is  to  be  re-arranged  in  the  upper  parts  and  transposed : 


Ex.  425. 


i8o 


GOODRICH'S  ANALYTICAL  HARMONY. 


At  (a)  two  other  arrangements  are  necessary,  (c)  is  a  re-arrange 
ment  of  (b),  therefore  one  other  arrangement  of  this  is  to  be  made. 
All  the  others  require  two  more  arrangements.  At  (b)  and  (c)  the 
5th  of  the  essential  discord  is  omitted  ;  at  (d)  the  3d  is  omitted.  At 
(e),  (f )  and  (g)  the  essential  discord  appears  in  its  different  inver- 
sions, the  positions  of  the  diminished  chord  being  governed  by  the 
former. 

One  circumstance  remains  to  be  explained  :  the  manner  in  which 
to  arrive  at  the  diminished  harmony.  In  the  last  example  this  is 
preceded  and  succeeded  by  the  essential  discord  to  avoid  presenting 
two  subjects  simultaneously.  Understand  that  the  progression  from 
the  antecedent  to  the  diminished  chord  must  be  theoretically  correct ; 
and  as  the  latter  is  not  a  fundamental  harmony,  a  melodic  progres- 
sion of  the  parts  is  preferable  to  skips.  The  student  should  become 
thoroughly  familiar  with  these  exercises  as  they  will  serve  all  ordi- 
nary purposes : 
Ex.  426. 


H=3F=?3::5=7 
iE?3EE*Ee 


IM 


5EE* 


ii 


The  diminished  chord  is  preceded  by  five  different  concords  and  by 
one  secondary  yth  in  order  to  give  the  base  a  melodic  progression. 
(This  is  necessary  here,  since  no  melody  appears  above.)  The  first 
four  examples  are  sufficiently  effective ;  (e)  is  the  least  desirable. 

The  base  may  occasionally  skip  to  some  omitted  note  of  the  di*nin- 
ished  chord,  as  here  : 


Ex.  427. 


P|~* 


instrumentally  these  skips  present  no  difficulties,  but  vocalists  not 
thoroughly  trained  are  liable  to  sing  the  altered  intervals  untrue. 

The  student  will  derive  much  benefit  from  writing  and  performing 
this  exercise  in  every  major  key,  as  it  contains  all  the  positions  to  be 
used  in  future  examples : 


GOODRICH  S    ANALYTICAL    HARMONY. 


181 


Ex.  428. 


This  is  sufficiently  varied  to  give  one  a  good  understanding  of  the 
use  of  these  chords. 

The  following  theme  when  harmonized  will  afford  a  practical  illus- 
tration of  the  theories  contained  in  this  and  a  previous  chapter.    The 
diminished  chords  resolved  secondarily  to  tonic  major  and  to  the 
-dominant  7th  are  to  be  employed : 
Ex.  429. 


Use  the  first  diminished  7th  chord  in  such  manner  as  to  leave  the 
tonic  chord  complete  in  the  upper  parts.  The  dashes  indicate  avoided 
cadences.  In  these  cases  the  essential  7th  chord  must  not  contain 
the  root  in  the  treble  parts,  and  this  will  necessarily  regulate  the  posi- 
tion of  the  enharmonic  discord.  The  second  measure  may  be  written 
as  at  (a)  or  (b)  : 


Ex.  430. 


Transpose  into  E-flat  and  F. 

The  diminished  7th  chord  is  sometimes  used  as  a  passing  har- 
mony to  a  minor  chord,  especially  when  it  leads  to  the  second  inver- 
sion of  the  latter. 

Example  from  a  Fantasia  by  Emanuel  Bach : 


-i,      F~F—  ^—     :r— i — di 

1    & 


t.-flat  here  takes  the  place  of  d-sharp  used  in  C-major,  because  e-flal 
is  the  natural  minor  3d.     The  key-note  remains  as  connecting  link; 


182 


GOODRICH  S    ANALYTICAL    HARMONY. 


f-sharp  ascends  to  g  and  a-natural  resolves  down  to  g,  as  heretofore. 
This  latter  is  the  weakest  feature  of  the  resolution,  for  the  sharpened 
6th  is  generally  foreign  to  the  harmony  of  a  minor  scale,  especially 
in  descending.     Mozart  employs  it  to  better  advantage  : 
Ex.  432. 


The  diminished  chord  results  from  the  chromatic  progression  below, 
following  the  upper  parts,  which  brings  the  3d  in  the  base.  Beet- 
hoven used  this  chord  in  similar  manner.  A  comparison  of  the  major 
and  minor  resolutions  will  show  to  better  advantage : 


Ex.  433. 


The  a-sharp  at  (a)  becomes  b-flat  at  (b).  G  remains,  e  descends  to 
d  in  both  instances,  and  c-sharp  ascends  to  d,  as  the  dominant  is  the 
same  in  both  modes.  The  first  is  more  natural  unless  the  minor  6th 
is  retained  in  the  second  illustration  : 


etc. 


Ex.  434. 


.But  this  is  no  longer  a  principal  diminished  yth  chord,  as  it  has  a 
diminished  3d.     (See  Ex.  524,  first  measure.) 


GOODRICH'S  ANALYTICAL  HARMONY. 


183 


Chapter  XLI. 


HARMONIC  CADENCES  IN  MAJOR. 

i.  Authentic.     2.  Complete.     3.  Perfect.     4.  Extended- 
Perfect.      5.  Avoided.     6.  Deceptive.     7.  In- 
complete (Half).     8.  After  (Plagal). 


THE  musical  definition  of  cadence  is  close,  termination,  or  ending. 
A  cadence  may  be  intermediate  or  final,  complete  or  incomplete. 

i.  AUTHENTIC  CADENCE. 

This  has  been  described  as  an  ending  by  means  of  the  harmonies 
of  the  dominant,  or  dominant  yth,  followed  by  the  tonic : 


Ex.  435. 


fcjh*=f-p^=g: 

m-tt^Z^ 


1 


These  may  be  written  in  any  position.    The  effect  is  satisfactory,  and 
they  are  therefore  suitable  for  the  ending  of  a  section  or  a  period. 

The  diminished  yth  chord  on  the  leading-note  is  frequently  em- 
ployed for  an  authentic  cadence  in  major,  though  it  is  more  natural 
to  the  minor  mode.  See  illustration  : 


Ex.  436. 


The  root,  5th  and  yth  of  the  discord,  each  resolving  up  or  down  a 


*  Hauptmann  considers  this  a  "  major-minor  key."    Such  dual  relationship  is  of  rare 
occurrence. 


184 


GOODRICH'S  ANALYTICAL  HARMONY. 


minor  2d,  impart  to  this  cadence  an  air  of  seriousness  not  observable 
in  the  minor  mode. 

In  resolving  this  discord  to  major,  the  5th  is  sometimes  given  the 
base  and  made  to  descend  a  4th.  Rossini  has  resolved  it  in  this  way 
in  the  chorus  parts  of  Inflammatus. 

2.  COMPLETE  CADENCE. 
This  consists  of  the  subdominant,  dominant,  and  tonic  harmonies : 

-tf—    i        i ^^ 


Ex.  437. 


— zy- 


The  complete  cadence  thoroughly  establishes  the  tonality,  containing 
as  it  does  every  tone  in  the  scale.  These  are  sufficient  to  form  a  har- 
monic cadence-group,  and  hundreds  of  melodies  have  been  harmo- 
nized with  the  three  chords.  (See  Ground-Base.) 

Another  form  of  this  cadence  consists  in  substituting  the  relative 
minor  of  the  subdominant  for  the  latter.  As  there  is  a  difference  of 
but  one  tone  between  these  two  harmonies  they  create  very  nearly 
the  same  impression.  Compare  this  with  Ex.  437. 


Ex.  438. 


In  the  first  cadence  the  3d  of  the  minor  triad  is  placed  in  the  base, 
because  c  is  the  subdominant.  In  the  second  cadence  all  the  chords 
are  uninverted.  The  former  is  of  more  frequent  occurrence.  An- 
other form  is  to  include  the  yth  of  the  dominant  chord : 


Ex.  439. 


V  *    «- 

§' 

i 

JL_        '?. 

(2. 

<n    • 

fm       2: 

—  zi 

%    • 

l>  \) 

~-~^ 

* 

\          \ 

f^V*  ** 

i          \ 

es    •        i 

, 

~£  ^  ^~~ 

1 

(1) 


GOODRICH'S  ANALYTICAL  HARMONY. 


185 


But  this  is  not  materially  different  from  the  others.  The  dominant 
chord  makes  the  actual  close  in  this  as  in  the  authentic  cadence, 
though  the  former  is  more  comprehensive. 

3.   PERFECT  CADENCE. 

This  includes  the  same  harmonies  as  the  preceding,  with  the  addi- 
tion of  the  second  inversion  of  the  tonic  chord  between  the  subdorni- 
nant  and  dominant,  thus : 


Ex.  440. 


-fl-i 

^^ 

*    ^ 

s 

5-  —  -  5 

E52 

(M)       2* 

__ 

<?     • 

F* 

^-*^ 

g       5 

P«*     ^5 

1 

B 

B 

BJ-             G^ 

With  the  connecting-note  throughout,  the  effect  is  more  smooth  than 
the  direct  progression  from  subdominant  to  dominant. 

Observe  that  the  chord  marked  (2)  is  connected  with  its  antecedent 
and  consequent. 

The  other  two  positions  with  the  same  base  are  to  be  written. 


PRACTICAL   EXERCISES. 

The  perfect  cadence  is  so  comprehensive  that  the  author  advises 
students  to  perform  it  without  notes,  in  all  major  scales,  after  having 
written  it  theoretically. 

There  are  important  advantages  to  be  derived  from  this  practice  : 

1.  It  familiarizes  one  with  the  most  important  fundamental  har- 
monies as  they  naturally  follow  one  another  in  any  particular  key, 
and  enables  one  to  anticipate  the  harmonic  basis  of  a  considerable 
quantity  of  modern  music. 

2.  By  performing  these  cadences  as  directed,  listening  attentively 
to  the  effect  of  each  chord,  the  hearing  faculties  will  gradually  become 
cultivated.     All  these  cadences  and  harmonizations  are  to  be  judged 
principally  by  the  sense  of  hearing  ;  but  if  that  sense  remains  uncul- 
tivated the  student  will  have  no  means  of  distinguishing  between 
good  and  bad  effects. 

3.  The  act  of  performing  all  cadences  and  harmonic  progressions, 
after  having  written  them  correctly,  serves  to  re-inforce  theoretical 


1 86 


GOODRICH'S  ANALYTICAL,  HARMONY. 


knowledge,  and  furnishes  what  is  most  needed  to  make  abstract  for- 
mulas serviceable.  Theory  is  valueless  unless  it  can  be  practically 
applied. 

A  model  for  transposition  is  presented  : 

Ex.  441. 


=£ 


I    i p  £    & 


I 1   ^          ' 


^ 


m 


•% 


?=:^ 


mp 


-IS 


i 


The  ties  show  the  connections  according  to  the  principles  of  chord 
progression. 

By  listening  to  these  cadences  the  student  will  eventually  learn  to 
hear  the  effect  of  a  progression  as  soon  as  it  is  seen  on  paper.  The 
author  considers  this  practice  of  the  highest  importance,  and  hopes 
it  will  no  longer  be  neglected. 


Another  form  of  this  cadence  consists  in  using  an  inverted  second- 
ary jth  chord  in  place  of  the  subdominant,  the  base  being  the  same : 


Ex.  442  7. 


Re-arrange  and  transpose  this.     The  secondary  discord  at  +   is  a 
combination  of  the  subdominant  harmony  and  its  relative  minor : 


It  is  therefore  well  adapted  to  the  cadence,  especially  when  its  3d  is 
in  the  base. 

*  These  ending-s  with  the  3d  or  sth  uppermost  are  here  considered  equally  satisfactory 
Vith  the  one  at  (b.)  Some  part  must  sound  the  3d  and  the  sth,  and  these  may  fall  to  the 
soprano,  as  well  as  to  the  contralto  or  tenor. 


GOODRICH'S  ANALYTICAL  HARMONY. 


187 


GROUND    BASE. 

An  illustration  of  this  cadence  repeated  three  times  identically,  as 
a  Ground-Base,  is  cited  from  the  Overture  to  Zampa  : 

Ex.  443. 
.Allegro 


Herold. 


II 


The  cadence-harmonies  at  (a)  are  repeated  literally  at  (b)  and  at  (c). 
Meanwhile  the  scale  passages  on  the  violins  present  a  constantly 
changing  melodic  design  which  rests  upon  the  ground-base  below. 
At  an  earlier  period  pieces  called  Grounds  were  much  in  vogue. 
These  consisted  of  the  complete  or  perfect  cadence-harmonies,  either 
in  major  or  minor,  as  a  ground  work.  The  melody  was  so  devised 
that  the  group  of  repeated  chords  would  serve  as  accompaniment. 
Though  little  used  at  the  present  time,  the  Ground  is  an  interesting 
study. 

By  changing  the  secondary  yth  chord  into  a  diminished  harmony 
a  more  gradual  progression  may  be  introduced : 


Ex.  444. 


i88 


GOODRICH'S  ANALYTICAL  HARMONY. 


This  +  indicates  the  passing  diminished  7th  chord,  which  is  very 
euphonious  and  has  been  much  used.  The  secondary  discord  might 
be  omitted  but  this  would  neither  be  so  smooth,  nor  so  progressive. 
In  both  instances  the  tonic  is  the  connecting  link,  before  and  after 
the  chromatic  chord. 

Re-arrange  and  transpose  all  examples. 

4.     EXTENDED-PERFECT  CADENCE. 

This  embraces  an  additional  harmony.  The  chord  of  the  relative 
minor  is  most  frequently  used  in  this  cadence,  and  comes  between 
the  tonic  and  subdominant,  being  closely  connected  with  both : 


Ex.  445. 


By  prolonging  the  value  of  each  chord  this  cadence  might  easily  em- 
brace an  entire  period.  It  also  gives  to  the  base  part  a  more  inde- 
pendent melody. 

In  place  of  the  diminished  chord,  the  relative  minor  with  its  root 
in  the  base  could  be  introduced ;  or  the  sequence  of  descending  thirds 
in  the  fundamental  part  might  be  continued  in  this  manner : 


Ex.  446. 


y      J      ^ 
l—& 


m 


The  cadence  may  also  be  extended  by  repeating  certain  chords  in  an 
inverted  form. 

(This  should  be  attempted  by  the  student.) 

5.     AVOIDED  CADENCE. 

The  application  and  effect  of  avoided  cadences  were  explained  in 
Chapter  XXV.  They  take  place  whenever  a  principal  discord  re- 
solves to  some  other  concord  than  that  to  which  it  naturally  belongs. 
The  positions  most  favorable  for  this  purpose  are  these : 


GOODRICH  S    ANALYTICAL    HARMONY. 


Ex.  447. 


11 rs H^ 1  — fjf -y- 1  —  <-j> '. 

ff Us P* 1-  — -^ ^ 1 x5~ 1 

^ <S— I -^ &—*-&          & "I 

1 , I-    pi s — I—  P* *> 1 

i L. 1 — , 1 1 — . 1 1 


Here  the  E-minor  chord  takes  the  place  of  the  tonic  chord  (G)  at  the 
natural  termination  of  a  melodic  idea.  It  preserves  the  interest  and 
prolongs  the  period,  for  this  is  an  indirect  resolution. 

An  example  is  given  showing  the  situation  in  which  an  avoided 
cadence  produces  the  best  effect : 


Ex.  448. 


The  dash  shows  where  the  cadence  is  avoided,  this  being  the  3d  reso- 
lution of  the  essential  discord.  (The  fourth  resolution,  corresponding 
in  minor  to  this,  will  appear  in  the  next  chapter.)  No  satisfactory 
close  occurs  until  the  appearance  of  the  tonic  chord  marked  /TV  The 
last  example  does  not  admit  much  re-arrangement,  but  should  be 
transposed.* 

6.     DECEPTIVE  CADENCE. 

The  principal  difference  between  avoided  and  deceptive  cadences 
is  that  in  the  latter  discord  follows  discord.  Reference  is  here  made 
to  the  last  of  a  period  where  the  final  dominant  yth  chord  would 
naturally  be  expected  to  end  on  the  tonic  chord.  But  if  another  dis- 
cord be  substituted  for  the  tonic  triad,  a  deceptive  cadence  occurs, 
and  the  music  must  continue  until  a  satisfactory  close  is  reached. 
Following  is  an  example : 


(No.  11,  Peters'  Ed.) 


Allegro. 


Sonata,  Haydn. 

M 


Ex.  449. 


*  The  Andante  to  Schubert's  B-flat  Symphony  contains  excellent  illustrations  of  avoided 
cadences  ;  also  the  Alia  Marcia  in  Schumann's  Op.  44. 


190 


GOODRICH'S  ANALYTICAL  HARMONY. 


This  is  the  last  of  an  eight-measure  period  ending  naturally  at  8. 
Here  the  deceptive  cadence  has  the  effect  of  considerably  extending 
the  actual  period,  until  the  essential  discord  resolves  io  the  tonic 
triad. 

A  similar  instance  is  presented,  to  which  the  same  remarks  will 
apply : 


Ex.  450. 


^ 

0. 


#"" 

The  last  measure  is  a  deceptive  cadence.* 

An  interesting  instance  may  be  found  in  the  Largo  of  Beethoven's 
Op.  7,  where  the  regular  close  is  prolonged  from  the  2oth  to  the  24th 
measure. 

7.  INCOMPLETE,  OR  HALF  CADENCE. 

The  incomplete,  or  half  cadence,  consists  of  the  tonic  followed  by 
the  dominant  harmony,  with  the  5th  of  the  former  in  the  base. 
This  naturally  leads  to  the  dominant : 


Ex.  451. 


2) 


The  tonic  chord  is  needed  to  form  a  perfect  cadence.  It  is  therefore 
incomplete,  and  presupposes  that  something  else  follows.  Illustra- 
tion (b)  has  the  same  effect  of  keeping  in  abeyance  the  actual  cadence 
and  is  lacking  in  that  sense  of  repose  and  completeness  which  only 
the  tonic  can  impart.  After  the  pause,  G-major,  G-minor,  D-major, 
D-minor,  or  even  other  keys  may  be  introduced. 

The  Sonatas  of  Haydn,  Mozart,  dementi,  Dussek,  and  the  earlier 
ones  of  Beethoven,  contain  many  instances  of  the  incomplete  cadence, 
which  usually  occurs  at  the  end  of  the  principal  theme. 


"To  distinguish  these  intermediate  cadences  the  reader  must  understand  the  analytical 
divisions  of  a  period  and  where  the  melodic  cadences  occur.     Musical  Analysis  explains  this. 


GOODRICH'S  ANALYTICAL  HARMONY. 


8.     AFTER  CADENCE   (PLAGAL). 


291 


The  subdominant  harmony  followed  by  that  of  the  tonic  consti- 
tutes what  is  known  as  a  plagal  cadence.  The  author  calls  it  an 
after  cadence  as  this  term  is  more  significant ;  this  cadence  coming 
after  the  final  ending  of  a  composition,  to  which  it  serves  as  a  short 
coda  or  extension.* 

In  church  music  the  after  cadence  is  often  used  at  the  end  of  an 
Anthem  or  Te  Deum  as  accompaniment  to  the  word  Amen.  Hence 
it  is  frequently  called  the  Amen  Cadence.  In  instrumental  music 
the  application  is  the  same  : 

-4-^- 


Ex.  452. 


A  -  men. 


The  phrase  in  brackets  shows  the  application  and  effect  of  the  after 
cadence.  The  period  terminates  before  this,  on  the  G  chord.  This 
cadence  may  be  written  in  various  ways,  thus : 


Ex.453. 


m 


At  (b)  the  passing  note  between  e  and  d  is  included.  At  (cv  the  sub- 
dominant  minor  does  not  appear  till  the  last  of  the  measure,  v»ut  the 
chromatic  progressions  in  the  middle  parts  naturally  lead  to  th^  sub- 
dominant  and  tonic. 

Of  all  the  cadences  herein  enumerated  the  after  cadence  is  the 
most  mild  and  undecided.  The  diminished  /th  chord  as  a  passing 
harmony  may  be  included  among  the  Amen  cadences : 


Ex.  454. 


*The  use  of  this  cadence  as  a  substitute  for  the  dominant  in  o!  !  ecclesiastic  music  is  inw 
obsolete,  and  the  Greek  term  has  no  real  significance  in  modern  music. 


IC)2 


GOODRICH  S    ANALYTICAL    HARMONY. 


The  tonality  of  G  is  not  affected  by  the  chromatic  passing  chord 
especially  as  the  tonic  remains  above  and  below.  The  harmony  oi 
a  major  3d  below  may  also  be  included,  though  its  relationship  i: 
apparently  remote : 


Ex.  455. 


The  effect  is  somewhat  transitory  and  unexpected,  but,  like  the  others 
it  serves  a  particular  purpose.  This  is  beautifully  illustrated  in  tin 
song  by  Kiicken,  Good  night,  farewell. 


Chapter  XLII. 


HARMONIC  CADENCES  IN  MINOR. 

i.  Authentic.     2.  Complete.     3.  Perfect.     4.  Extended- 
Perfect.     5.  Avoided.     6.  Deceptive.     7.  In- 
complete.    8.  After.     9.  Ambiguous. 

i.  AUTHENTIC   CADENCE. 

THE  principal  cadences  in  the  minor  mode  are  based  upon  th< 
same  fundamentals  that  were  employed  in  major.     And  as  th< 
dominant  harmony  is  identical  in  both  modes  the  auvhentic  cadenc* 
will  consist  of  these  chords,  in  any  position  : 


Ex.  456. 


GOODRICH'S  ANALYTICAL  HARMONY. 


193 


Compare  this  with  Ex.  435.     Every  note  of  the  minor  scale  is  here 
employed  except  the  6th. 

The  diminished  yth  chord  plays  a  more  important  part  here  than 
in  the  major  cadences,  for  the  discord  and  its  principal  resolution 
embrace  every  note  in  the  harmonic  minor  scale : 


Ex.  457. 


This  principal  resolution  of  a  diminished  7th  chord  decides  the  minor 
key  as  satisfactorily  as  does  the  dominant  yth  harmony.  For  termi- 
nal resolutions  of  this  discord  the  student  will  dp  well  to  use  only 
these  positions : 


Ex.  458. 


As  the  tonic  appears  in  the  base  after  each  resolution,  they  are  all 
final.  Of  the  illustrations  (a)  and  (b)  are  best.  At  (c)  the  7th  ap- 
pears uppermost  and  this  necessitates  the  duplicating  of  the  3d  to 
prevent  fifths.  If  it  is  desirable  to  have  the  last  chord  complete  in 
the  upper  parts,  double  the  root  and  omit  the  3d  or  5th. 

When  the  3d  appears  as  real-base  it  resolves  intermediately  (a), 
and  the  corresponding  dominant  7th  will  be  necessary  for  the  final 
cadence  (b)  : 


Ex.  459. 


(1)  (2) 


So  with  the  other  inversions.     This  is  the  only  objection  to  the  di- 
minished 7th  chord  in  a  final  cadence. 

Remember  that  the  root,  5th  and  7th  have  fixed  resolutions,  while 
the  3d  may  resolve  according  to  circumstances. 


194 


GOODRICH'S  ANALYTICAL  HARMONY. 
2.  COMPLETE  CADENCE. 


Here  the  tonic  and  subdominant  are  minor,  while  the  domman 
remains  major. 

Complete  cadences  in  major  and  minor  follow : 


Ex.  460. 


Observe  that  the  fundamentals  are  identical. 

The  complete  cadence  embraces  all  tones  of  the  harmonic  mino 
scale. 

The  cadence  given  at  (b)  admits  of  re-arrangement  according  t< 
this  model,  and  must  also  be  transposed.  *  *  * 

The  second  form  of  this  cadence  given  in  major  does  not  admi 
of  exact  reproduction,  because  the  subdominant  is  itself  minor,  an< 
therefore  has  no  relative  minor.  But  a  form  may  be  used  corres 
ponding  to  that  given  in  Ex.  438,  by  substituting  an  imperfect  tria< 
on  the  supertonic  for  the  subdominant  harmony : 


Ex.  461. 


(1) 

The  second  chord  contains  two  notes  common  to  the  subdominan 
harmony,  and  by  placing  the  3d  in  the  base  a  very  good  substitut( 
for  the  harmony  of  the  4th  is  obtained.  On  account  of  the  inverted 
triad  it  is  better  to  move  the  parts  contrarily,  as  at  (a),  (b)  and  (c) 
Contrary  movement  is  also  better  in  going  from  the  first  to  the  sec 
ond  chord,  because  there  is  no  connecting  note.  Transpose  these 


*This  affords  another  argument  in  favor  of  the  harmonic  minor  scale. 


GOODRICH'S  ANALYTICAL  HARMONY. 
3.  PERFECT  CADENCE. 


195 


ia  io  reptodd<_ed  by  using  the  same  harmonies  according  to  the 
tonality : 


Ex.  462. 


The  dominant  chord  at  the  close  ii  sufficient,  but  the  7th  is  included 
in  order  to  follow  the  downward  tendency  of  the  theme.  Should  the 
melody  suggest  it  the  imperfect  triad  may  be  used  in  place  of  the 
subdominant.  Otherwise  it  would  be  the  same  as  Ex.  462  : 


Ex.  463. 


Sf 


h* 


e^E 


(i) 

In  a  final  close  this  might  be  preferable  to  the  essential  discord  as  a 
harmonization  of  the  melody  note  at  + . 

The  subdominant  harmony  can  be  combined  with  the  imperfect 
triad,  making  a  secondary  yth  somewhat  similar  to  the  one  employed 
in  major: 

r^         t 

^ 


Ex.  464. 


-•j 


(1)     (2) 


:£= 


B 


This  corresponds  to  Ex.  442  (a).    The  3d  is  usually  treated^s  a  real- 
base  in  order  to  produce  the  effect  of  a  subdominant  harmony. 

There  is  a  foreign  harmony  frequently  employed  in  a  minor  ca- 
dence that  deserves  a  place  here.  Counting  from  the  fourth  as  real- 
base  it  contains  a  small  3d  and  normal  4th.  Theoretically  it  is  a 
major  chord  located  a  minor  2d  above  the  key-tone ;  in  actual  prac- 
tice it  is  treated  as  a  derived  harmony,  like  the  augmented  6th  chords. 
It  is  known  as  the  "  Neapolitan  Sixth,"  but  this  appellation  does  not 


196 


GOODRICH'S  ANALYTICAL  HARMONY. 


seem  to  be  appropriate.  Every  modern  composer  has  used  it  with- 
out regard  to  local  origin  or  association.  In  his  Italian  symphony 
Mendelssohn  does  not  employ  it,  but  in  the  Scotch  symphony  it 
occurs  several  times.  This  would  seem  to  prove  that  no  local  color- 
ing is  to  be  derived  from  this  cadence,  for  Mendelssohn  was  quick  to- 
avail  himself  of  any  extraneous  aid  to  legitimate  tone  painting.  Ros- 
sini uses  the  cadence  in  his  Swiss  overture,  as  well  as  in  his  Neapol- 
itan tarantella,  La  Danza : 


BESS^^BE 

1>W      4      *             •                      1 

5 

2_ 

X9      •                  U^5      • 

V           1            1 

*    € 

x  -J.  J. 

*    a 

) 

)•  .     ?«.'           lr?        •                     \       *S        • 

jO        * 

/^  •(* 

•S  rt    i't      i                   [ZSZIj 

9jJ 

1 

f      4:          U       .                   1      f 

i 

Ex.  465. 


This  so-called  Neapolitan  6th  furnishes  a  major  harmony  with  the 
subdominant  as  real-base,  in  place  of  the  imperfect  triad.  It  is  much 
brighter  than  the  latter,  and  tends  to  relieve  the  sombre  hues  of  the 
harmonic  minor  scale.  Transpose  the  example  into  several  other 
scales.  The  passing  diminished  yth  employed  in  Ex.  444  has  no 
equivalent  in  the  minor  mode,  because  there  is  no  recognized  tone 
between  2  and  3  of  the  scale.  But  4  in  the  base  might  be  sharpened 
as  a  passing-note  to  5.  See  Chapter  XL. 

4.  EXTENDED-PERFECT  CADENCE. 

This  may  progress  by  thirds  as  in  major.     It  is  only  necessary  to 
follow  the  signature : 


Ex.  466. 


Either  form  may  be  utilized. 

5.  AVOIDED  CADENCE. 

The  same  general  principles  apply  to  both  modes.    The  example 
is  therefore  presented : 


GOODRICH'S  ANALYTICAL  HARMONY. 


197 


Ex.  467. 


The  fourth  resolution  corresponds  to  the  third  in  major.  These  are 
employed  whenever  it  is  desirable  to  prolong  the  cadence.  The  pro- 
gression from  f-sharp  down  to  e-flat  in  the  middle  of  the  example  is 
perfectly  correct,  notwithstanding  the  attempted  prohibition  of  this 

augmented  2d. 

6.  DECEPTIVE  CADENCE. 

This  may  occur  in  the  minor,  as  well  as  in  the  major  mode.  The 
principle  and  the  general  effect  remain  the  same.  Examples  in  major 
are,  however,  more  common. 

7.  INCOMPLETE  CADENCE. 

By  comparing  the  next  example  with  No.  451,  the  student  will 
perceive  the  general  similarity  : 


Ex.  468. 


In  either  mode  it  is  an  incomplete  cadence. 

8.  AFTER  CADENCE  (AMEN). 

This  consists  of  the  subdominant  (or  some  harmony  correspond- 
ing to  that  of  the  fourth)  followed  by  the  tonic,  as  in  major.  With 
exception  of  the  difference  in  mode  the  effects  are  identical.  There 
is,  however,  this  important  distinction  to  be  made:  In  major  use 
either  a  major  or  minor  chord  on  the  subdominant ;  but  in  a  minor 
key  the  subdominant  harmony  is  naturally  minor:* 


Ex.  469. 


Good. 


bad. 


e) 


*  In  the  Sicilienne  from  "  Cavalleria  Rusticana  "  this  natural  order  Is  reversed  and  with 
excellent  effect. 


1 98 


GOODRICH'S  ANALYTICAL  HARMONY. 


The  major  chord  in  the  second  example  is  usually  ineffective.  The 
situation  in  which  the  amen  cadence  occurs  was  illustrated  in  the 
previous  chapter.  The  harmony  of  a  major  3d  below  the  tonic  might 
be  used  as  a  modification  of  the  after  cadence,  and  in  a  minor  scale 
the  effect  would  be  less  abrupt  than  in  major.  The  diminished  chord 
is  not  available  as  it  was  in  major. 

g.  AMBIGUOUS  CADENCE. 

In  Chapter  XLJX  this  cadence  occurs  among  the  harmonies  ot 
the  natural  minor  scale.  It  consists  of  the  dominant  minor,  in  place 
of  dominant  major,  with  the  subtonic  as  melody  note.  This  is  fol- 
lowed by  the  tonic  harmony  : 


Ex.  470. 


It  is  rather  weak  and  melancholy,  and  should  be  used  in  accordance 
with  these  sentiments.  Its  effect  is  more  satisfactory  if  not  brought 
into  immediate  comparison  with  the  more  positive  and  familiar  dom- 
inant major  harmony.  An  instance  is  here  quoted  from  Grieg,  whom 
the  author  considers  the  greatest  harmonist  now  living : 


Ex.47i. 


n     p  tt»       '  '  »                     ^ 

^  ^-^^    Op.  22. 

V         I      *T       1*         1        ^                      F—  " 

•^    ^»        <*               1 

y2_      j  

:  ~^J                 -1 

Rit. 

n  •           "^" 

i*F          "a 

*  *            & 

")• 

ft                       "<• 

^                                               q 

n    -                   »\ 

PP 

~ 

c^* 

i 

^'    g  

f—      -^-" 

The  composer  evidently  anticipated  that  some  academic  musician 
would  "  correct "  this  cadence,  for  the  natural  is  found  before  d  in 
both  parts  of  the  duet.  This  would  be  less  effective  in  the  majoi 
mode. 


''All  these  cadences  should  be  performed  iu  various  minor  scales. 


GOODRICH'S  ANALYTICAL  HARMONY. 


199 


PART  X. 


Chapter  XLIII. 


AUGMENTED  SIXTH  CHORDS— THEIR  DERIVA- 
TION, APPLICATION  AND  EFFECT. 

No.  i. 

A  MAJOR  6th  enlarged,  becomes  what  is  known  as  an  "  extreme 
sharp,"  or  augmented  6th.     It  is  two  chromatic  tones  larger 
than  a  minor  6th,  and  one  chromatic  tone  larger  than  a  major  6th. 
In  notation,  thus: 

Min.      Maj.      Aug.       Aug. 

• l-i -T 

Ex.  472. 


The  augmented  6th  may  be  produced  by  raising  the  upper,  or  low- 
ering the  lower  tone  of  a  major  6th,  as  illustrated.  In  both  instances 
the  direct  resolutio-n  is  to  the  octave  situated  a  minor  2d  above  and 
below  the  interval  of  the  augmented  6th  : 


Ex.  473. 


This  interval  may  appear  as  a  i3th  or  a  2oth  from  the  real-base  (c  or 
c-Jlat  in  last  example).  It  is  still  called  by  the  same  name  and  treated 
hi  the  same  manner  as  though  the  interval  of  an  augmented  6th 
appeared  uninverted ;  for  it  is  this  interval  that  gives  the  chord  its 
name,  independently  of  the  intermediate  tones  omitted  from  the  last 
example. 


2OO 


GOODRICH'S  ANALYTICAL  HARMONY. 


There  are  various  chords  containing  the  interval  of  an  augmentet 
6th.  The  author  has  systemized  these,  and  endeavored  to  explaii 
them  as  they  are  used  by  standard  composers.  They  are  numberec 
from  i  to  3.  No.  i,  with  its  most  natural  derivation,  is  given  first 
Begin  with  a  yth  chord,  species  IV,  as  at  (a),  invert  it  once  (b)  am 
raise  the  original  root,  now  the  6th,  (c) : 


Ex.  474. 


No.  1. 


The  theoretical  derivation  of  the  augmented  6th  chord,  No.  i  (c),  i 
here  shown  as  a  formula.  In  actual  practice  the  original  positioi 
(a)  seldom  appears,  and  it  is  unnecessary  to  go  through  this  process 
From  the  real-base  (/)  to  the  d-sharp  is  an  augmented  6th,  and  a 
this  chord,  to  be  most  characteristic,  must  have  its  3d  in  the  base 
the  propriety  of  calling  it  a  6th  chord  is  seen,  though  the  6th  (/t< 
</)  was  originally  a  3d,  dtof.  Compute  the  intervals  from  the  real 
base :  f  to  a,  a  major  3d ;  f  to  c,  a  normal  5th  ;  and/  to  d-sharp,  ai 
augmented  6th.  These  are  the  component  parts  of  the  chord  calle< 
No.  i.  Inasmuch  as  the  extreme  parts  (/"and  d-sharp)  must  resolv 
up  and  down  to  the  octave  it  should  be  considered  as  part  of  th 
resulting  concord : 


This  is  given  first  because  the  augmented  6th  has  a  fixed  resolution 
Suppose  the  two  middle  parts  were  to  remain  stationary,  would  the; 
form  a  concord  in  connection  with  el  If  so,  this  chord,  e,  a,  c,  is  t< 
be  considered  the  resolution.  The  d-sharp  would  seem  'to  indicat 
the  chord  of  E,  as  d-sharp  is  the  leading  note  to  E.  But  this  woul< 
involve  parallel  fifths  and  can  not  be  recommended : 


Ex.  476. 


The  minor  triad  must  therefore  be  used  in  its  third  position,  5!.! 
below : 

Ex.  477. 


GOODRICH'S  ANALYTICAL  HARMONY. 


20 1 


It  is  evidently  impossible  to  stop  on  this  inverted  triad.  The  domi- 
nant, or  dominant  7th  chord  founded  upon  the  tone  of  the  real-base 
</)  must  follow,  and  this  will  lead  naturally  to  the  cadence  upon  A : 


Ex.  478. 


This  constitutes  another  form  of  perfect  cadence,  and  another  method 
of  modulating  to  A-minor. 

The  entire  process  should  be  summed  up  in  this  way  :  The  aug- 
mented 6th  chord,  No.  i,  resolves  to  a  minor  chord  in  its  third  posi- 
tion ;  then  to  the  dominant  to  this,  and  finally  the  tonic. 

In  writing  different  positions  the  base  is  to  remain  the  same,  but 
the  other  parts  of  the  chord  may  be  freely  re-arranged,  and  resolved 
in  the  same  manner.  The  other  two  arrangements  are  given  as  a 
model : 


Ex.  479, 


The  three  examples  (a),  (b)  and  (c)  are  similar.  The  jth  of  the  domi- 
nant chord  may  be  included,  as  at  (b)  and  (d),  or  omitted,  as  at  (a) 
and  (c). 

It  is  important  to  observe  of  all  these  augmented  6th  chords  that 
the  tone  a  minor  2d  below  the  real-base  is  the  dominant  to  the  final 
tonic.  (Analyze  it  in  this  manner :  f  is  the  real-base  in  the  last  two 
examples;  a  minor  2d  below/ is  <?,  and  e  is  the  dominant  to  the  final 
tonic,  ./.) 

Next  select  the  same  kind  ot  a  secondary  jth  chord  (IV),  founded 
upon  the  third  of  the  original  scale.  Invert  this  once  (3d  in  the  base) 
and  sharpen  the  6th.  G  is  now  the  real-base,  and  this  contains  the 
same  theoretical  intervals  as  does  the  first  augmented  6th  chord  that 
was  produced;  i.  c.,  a  major  3d,  normal  5th,  and  augmented  6th  It 
is  therefore  numbered  and  resolved  in  the  same  manner.  This  should 


No.  1.  Third  arrangement 


2O2  GOODRICH'S  ANALYTICAL,  HARMONY. 

be  written  out  by  the  student  as  far  as  the  final  resolution  to  tonic, 
according  to  the  principles  herein  explained.     *     *     * 

This  last  chord  performs  a  transition  outside  of  the  circle  of  nearly 
related  keys ;  i.  e.,  from  C-major  to  B-minor.  Each  example  is  ar- 
ranged in  three  close  positions.  One  of  these  is  given  for  reference : 


Ex.  480. 


The  other  arrangements  should  correspond  in  treatment  to  this.  In 
writing  these  examples  use  both  forms,  with  and  without  the  jth, 
and  without  altering  the  base. 

Another  secondary  yth  chord  of  the  same  species  may  be  trans- 
formed  into  an  augmented  6th  chord,  No.  i,  in  exactly  the  same  man- 
ner. This  is  founded  upon  the  submediant  (or  supexlominant) : 


Ex. 


The  student  should  go  through  the  process  of  inversion,  chro- 
matic alteration,  resolution,  etc.,  as  already  described,  completing  the 
examples  in  three  positions.  The  real-base  is  treated  as  root  and  is 
not  to  be  duplicated  above.  The  intervals  of  the  augmented  6th 
chord  are  computed  from  this  base  note.  Transpose  into  several 
other  major  scales.  *  *  * 

Select  the  three  secondary  jth  chords  of  species  IV,  founded  upon 
2,  3,  and  6  of  any  major  scale  and  derive  an  augmented  6th  chord 
from  each.  Place  the  original  3d  below  as  real-base,  sharpen  the  6th. 
and  resolve  as  directed :  The  interval  of  an  augmented  6th  resolves 
up  and  down  to  the  octave ;  the  3d  and  5th  (always  counting  from 
the  actual  base  note  when  an  augmented  6th  chord  is  under  consid- 
eration) remain  stationary.  This  invariably  results  in  an  indirect 
resolution,  the  concord  appearing  with  its  5th  below.  In  all  such 
instances  the  base  remains  as  root  of  the  dominant  harmony,  and 
then  the  cadence  naturally  takes  place.  All  these  will  contain  the 
same  theoretical  intervals,  and  be  resolved  identically.  Therefore 
number  them  I.  *  *  * 


GOODRICH'S  ANALYTICAL  HARMONY.  203 

The  other  derivations  of  this  altered  6th  chord,  No.  i,  are  now 
presented.  As  its  notes  correspond  on  the  key-board  to  those  of  a 
dominant  yth  whose  root  is  the  same  as  the  real-base  of  the  6th  chord, 
one  may  be  changed  into  the  other  by  means  of  a  slight  enharmonic 
alteration.  Even  this  partial  metamorphosis  gives  to  the  resolution 
a  very  different  direction  and  termination.  Such  an  instance  is  here 
presented,  with  the  natural  resolution  of  each  chord : 


Ex.  482. 


Compare  (a)  with  (b).  One  resolves  naturally  to  F '-major,  the  other 
leads  to  E-minor.  Yet  the  only  difference  between  the  two  discords 
is  that  expressed  by  b-flat  and  a-sharp.  Therefore,  when  writing  in 
F-»iajor,  the  key  of  E-minor  can  easily  be  established  by  making  the 
enharmonic  change,  as  illustrated  in  this  example : 


Ex.  483. 


The  treatment  of  the  6th  chord  is  the  same,  whatever  may  be  its  der- 
ivation. The  student  should  change  the  following  essential  discords 
into  augmented  6th  chords,  No.  i,  by  writing  the  yth  of  the  former 
enharmonically.  Then  work  out  the  resolutions  to  a  final  cadence 
as  heretofore : 

Ex.  484. 


The  discords  upon  F,  G,  and  C  will  yield  the  same  results  as  the  6th 
chords  founded  upon  those  tones.  They  are  included  here  to  show 
the  different  derivations  and  the  possibilities  of  transition.  But  the 
discord  upon  B-flat,  last  example,  will  result  in  a  modulation  to  D-mi- 
nor,  after  the  a-flat  becomes  g-sharp,  and  this  was  not  included  in 
previous  exercises.  Therefore  this  should  be  worked  out  more  com- 
pletely, and  in  three  positions. 

The  augmented  6th  chord.  No.  i,  may  also  be  derived  from  any 
of  the  following  diminished  jth  chords : 


204 


GOODRICH'S  ANALYTICAL  HARMONY. 


Invert  them  once,  and  flatten  the  real-base : 


Ex.  486.  : 


These  lead  to  G-minor,  A-minor,  and  D-minor.  Write  out  the  fin 
in  three  positions,  using  two  staffs,  and  one  position  of  the  others  i 
order  to  indicate  their  application.  The  resolutions  remain  the  sam< 
Another  method  of  producing  the  augmented  6th  chord  consist 
in  lowering  the  upper  root  of  a  major  concord  a  diminished  3d,  whil 
the  other  three  tones  remain  passive  : 


Ex.  487. 


This  may  be  done  with  the  chords  of  the  tonic  and  dominant  in  th 
same  manner.  The  resulting  resolutions  are  the  same  as  those  alread 
written ;  but  the  student  may  work  out  these  from  the  key  of  E-flat 


Ex.  488. 


The  upper  voice-part  may  also  descend  by  minor  seconds  to  tl 
sharpened  6th,  as  here : 


Ex.  489. 


Work  out  each  of  these  to  the  final  tonic.  They  lead  to  F-mino, 
B-flat  minor,  and  C-minor.  The  augmented  6th  chord  appears  o 
the  last  of  each  measure,  and  the  treatment  from  this  point  is  n( 
different  from  what  has  been  explained.  In  the  re-arrangements  th 
melodic  progression  of  two  minor  seconds  will  appear  alternately  i 
the  various  upper  parts.  The  chromatic  signs  must  not  be  neglecte 
in  these  transitions. 


GOODRICH'S  ANALYTICAL  HARMONY. 


205 


Chapter  XLIV. 


AUGMENTED  SIXTH  CHORDS  CONTINUED. 


No.  2. 


THE  first  augmented  6th  chord  was  resolved  to  an  inverted  minor 
triad  to  prevent  parallel  fifths,  which  would  have  resulted  had 
the  6th  chord  gone  direct  to  the  dominant  major : 


Ex.  490. 


The  first  resolution  at  (a)  is  indirect,  and  the  dominant  chord  comes 
after  this.  At  (b)  the  6th  chord,  No.  i,  goes  direct  to  the  dominant 
chord  on  F-sharp.  No  objection  can  be  offered  to  the  resolution  of 
g,  b,  and  e-sharp  of  the  second  measure.  But  the  tenor  part  moving 
down  by  fifths  with  the  base  is  generally  objectionable.  If  the  c-sharp 
of  the  last  chord  could  be  anticipated,  the  progression  would  be  good, 
and  there  would  be  a  connecting  note,  thus  : 


Ex.  491. 


This  is  an  augmented  6th  chord,  No.  2,  and  the  example  is  free  from 
error.  Where  strength  and  boldness  are  desired  this  is  highly  effect- 
ive. The  derivations  and  final  resolutions  of  No.  2  are  now  given. 
It  may  be  derived  from  the  discord  No.  Ill,  by  using  the  second 
inversion,  and  then  raising  the  6th.  The  original  5th  will  become 
real-base.  The  resolution  is  to  a  major  chord  direct.  This  dominan  . 
chord  is  located  a  minor  2d  below  the  real-base  of  the  discord.  The 
final  cadence  is  to  the  tonic  a  normal  4th  above  the  dominant  chord. 


200 


GOODRICH'S  ANALYTICAL  HARMONY. 


The  constituent  elements  of  this  6th  chord,  No.  2,  are  :  a  major  3( 
augmented  4th  (in  place  of  the  normal  5th)  and  augmented  6th.  Th 
real-base  and  the  sharpened  6th  resolve  as  usual  to  the  octave ;  th 
3d  descends  a  minor  2d,  and  the  augmented  4th  remains  as  a  cor 
necting  note  to  become  5th  of  the  dominant  chord.  See  last  examph 
The  chord  that  follows  the  augmented  6th,  No.  2,  is  to  be  cor 
sidered  as  dominant,  not  as  tonic ;  for  the  key  of  the  second  chord  i 
too  far  removed  from  the  original  tonality.  But  if  the  second  chor 
is  considered  as  dominant,  the  tonic,  being  one  of  the  related  key; 
naturally  follows,  as  here : 


Ex.  492. 


The  reason  for  this  is,  that  if  we  consider  F-sharp  as  the  tonic,  an 
stop  at  (a),  the  ear  recognizes  a  strange  tone  in  the  real-base,  g  goin 
down  a  half  step  to  f-sharp.  A  minor  2d  above  any  tonic,  eithe 
major  or  minor,  is  not  natural  to  that  key.  It  is  a  foreign  eiemen 
and  therefore  we  do  not  recognize  the  second  chord  as  final  toni< 
but  as  dominant.* 

Observe  the  difference,  both  in  appearance  and  effect,  between  th 
augmented  6th  chord,  No.  2,  and  the  dominant  yth,  each  resolvin 
to  F-sharp  major : 


M  " 

ffs~ $gs     [^     Hfcg     i^     • 


Ex.  493. 


The  resolution  at  (b)  is  much  more  final  and  usually  more  satisfac 
tory  than  the  one  at  (a).  This  leads  eventually  to  B -minor,  of  whie 
the  second  chord  is  dominant.  But  the  resolution  at  (b)  is  complet 
in  itself,  and  may  rest  there. 

*  The  author  merely  explains  here  an  elementary  principle  of  transition  and  tonalit; 
lieyond  this  general  principle  there  is  a  psychological  force  that  may  overcome  the  mo! 
important  theoretical  laws.  This  will  be  explained  in  the  analysis  of  Grieg's  composition: 
Meanwhile  these  principles  are  to  be  considered  inviolable. 


GOODRICH  S    ANALYTICAL    HARMONY. 


207 


The  student  should  complete  the  different  arrangements  of  Ex. 
492  according  to  the  preceding  explanations.  The  arrangement  illus- 
trated in  Ex.  492  is  sometimes  employed  in  order  to  end  with  the 
key-tone  uppermost,  but  an  occasional  exercise  of  this  kind  will  be 
sufficient.  In  the  re-arrangements  the  real-base  remains  the  same. 

£      $      $ 

This  discord  may  also  be  produced  from  a  dominant  yth  chord  in 
its  second  inversion.  With  the  dominant  jth  to  the  tonic  it  is  only 
necessary  to  flatten  the  real-base,  thus : 


Ex.  494. 


This  resolves  first  to  the  D-major  chord  (now  considered  as  domi- 
nant), and  the  final  tonic  is  G. 

With  regard  to  the  mode  of  the  final  tonic  the  author  recommends 
the  student  to  be  governed  by  the  prevailing  signature.  Thus  the 
last  example  ends  naturally  in  G-major ;  whereas  the  previous  ex- 
ample leads  naturally  to  B -minor.  Write  out  the  remainder  of  Ex. 
494,  and  re-arrange  it  in  two  other  close  positions.  *  *  * 

An  augmented  6th  chord,  No.  2,  may  be  derived  from  the  second 
inversion  of  any  essential  yth  chord  by  flattening  the  5th  as  real-base. 
It  will  be  sufficient  to  present  one  complete  example,  as  they  are  all 
treated  similarly : 
Ex.  495. 


c^S      *—  fc« 

-?B-  —  BEE3i 

—  /i«fx  .  3)  1 

-^  Z?— 

-1  T— 

On  account  of  the  /B  the  E-major  chord  is  here  treated  as  a  domi- 
nant, this  being  a  modulation  to  A-inajor.  At  (c)  the  minor  jth  is 
introduced,  though  this  is  not  strictly  necessary.  The  other  essen- 
tial discords  to  be  written  out  and  changed  into  augmented  6th  chords 
are  founded  upon  2  and  3  of  the  major  scale.  Each  one  is  to  be  placed 
in  its  second  inversion.  The  real-base  is  then  lowered  a  chromatic 
step,  as  in  the  last  example.  Arrange  these  in  three  positions.  A> 
two  have  already  been  wrkten  (besides  the  one  founded  on  the  sec- 
ondary yth),  these  will  include  all  that  are  practicable  in  this  kev. 


208 


GOODRICH  a    AAALVTICAt,    HARMONY. 


The  most  feasible  derivations  of  the  augmented  6th  chord.  No, 
are  therefore  as  follows :  From  dominant  yth  chords  founded  up 
2,  3,  5,  and  6  of  any  major  scale,  and  from  the  secondary  discord 
the  yth  of  the  scale.  Each  of  these  is  to  be  inverted  twice,  and  1 
real-base  is  flattened  in  those  derived  from  essential  discords.  T 
6th  chord  derived  from  the  secondary  yth  on  the  leading  tone  requi: 
110  alteration  of  the  real-baae,  because  the  original  5th  is  imperfe 
and  this  produces  the  augmented  4th.  See  Ex.  493  (a). 

All  these  instances  result  in  the  same  kind  of  an  augmented  ( 
chord,  and  their  treatment  is  therefore  identical,  excepting  the  fi 
that  two  end  in  minor  and  three  in  major,  according  to  the  signatu 
With  exception  of  the  6th  chord  produced  from  the  inverted  doi 
nant  yth  chord  the  other  derivations  may  come  from  secondary  i 
well  as  from  principal)  yth  chords  on  the  ad,  3d,  and  6th  of  any  ma_ 
scale.  If  a  secondary  discord  be  chosen  it  will  be  necessary  to  ra 
the  6th  in  addition  to  the  lowering  of  the  real-base.  By  using 
inverted  dominant  yth  the  major  6th  will  be  ready  to  hand,  and  th 
by  lowering  the  real-base  the  augmented  6th,  No.  2,  will  result.  Th< 
are  the  only  differences  between  the  two  methods ;  the  general 
suits  are  identical. 

Another  derivation  is  here  presented : 

Gleason. 


Ex.  496. 


The  6th  chord,  No.  2,  is  preceded  by  an  altered  triad,  having  the  effc 
of  an  augmented  6th  chord.  The  derivation  is  perfectly  natural.  T 
final  chords  to  be  written  and  re-arranged  will  (in  the  key  of  D)  le 
to  D-major,  E-minor,  G-major,  A-major  and  B-minor.  Reduce  the 
to  numbers  and  they  will  apply  to  any  major  key  in  transposin 
They  ought  to  be  worked  out  in  at  least  four  other  major  scales ;  ai 
the  practical  performance  is  not  to  be  neglected,  for  one  is  a  nec< 
^ary  complement  to  the  other. 


/p. 

RTl       BZ 

i          i 

"  K 

«y        &         n*  •     »      "• 
etc. 

J        J.    j^! 

j  •     5    nzg 

1*          t-^ 

2J 

r    j   ^ 

GGODRICH'S  ANALYTICAL  HARMONY.  209 


Chapter  XLV. 


AUGMENTED  SIXTH  CHORDS  CONTINUED. 

No.  3. 

BY  selecting  an  augmented  6th  chord,  No.  2,  and  raising  the  aug- 
mented 4th  still  farther,  there  will  result  an  augmented  6th  chord 
of  the  third  species  : 

Ex.  497. 


This  contains  a  major  3d;  a  doubly-augmented  4th,*  and  an  aug- 
mented 6th.  Its  resolution  is  to  the  second  inversion  of  a  major 
chord.  The  real-base  and  the  augmented  6th  resolve  as  usual ;  the 
3d  remains  as  connecting  link ;  and  the  doubly-augmented  4th  as- 
cends a  minor  2d.  The  dominant,  or  dominant  7th,  follows  the  in- 
verted concord,  and  then  the  final  cadence  naturally  takes  place. 
Complete  the  exercise,  and  make  two  other  arrangements,  preserving 
the  same  base.  *  *  * 

As  the  resolution  of  No.  3  is  necessarily  to  a  major  chord,  the  first 
task  is  to  select  the  three  major  chords  that  occur  naturally  in  every 
scale.  These  represent  the  three  related  major  keys,  tonic,  subdomi- 
nant,  and  dominant.  The  augmented  6th  chords  are  located  a  large 
3d  below  each  of  these  key -tones.  Or,  to  be  more  explicit,  the  real- 
base  of  any  augmented  6th  chord  is  a  major  3d  below  the  final  tonic. 
The  three  discords  from  which  these  augmented  6th  chords  are  de- 
rived are  therefore  founded  upon  the  2d,  5th  and  6th  of  a  major  scale. 
These  are  inverted  twice,  and  altered  into  augmented  6th  chords.  In 
Ex.  497  the  chord  numbered  2  was  derived  from  the  7th  chord  on  the 
dominant  to  C.  Those  discords  founded  upon  the  2d  and  6th  of  the 
scale  may  at  first  be  principal  or  secondary. 

The  student  can  complete  the  task  of  writing  out  in  different  posi- 
tions the  two  remaining  altered  6th  chords.  The}7  end  in  C  and  in 
G,  and  are  transitions.  Together  with  the  one  ending  in  F  as  final 

*  The  doubly-augmented  4th  is  two  chromatic  steps  larger  than  the  normal  4th. 


210 


GOODRICH  S   ANALYTICAL    HARMON V 


tonic,  these  will  include  all  the  related  major  keys  in  the  scale  of  C. 
It  is  of  no  present  consequence  whether  the  student  flattens  the  real- 
base  before  sharpening  the  augmented  4th,  or  vice  versa.  *  *  * 
In  case  it  were  desirable  to  extend  the  progressions,  begin  with  a 
secondary  jth  chord  on  the  2d  or  6th  of  the  scale  and  gradually  altei 
the  intervals  until  an  augmented  6th,  No.  3,  appears : 


Ex.  498. 


II          3 

=,2=tez 


£ 


i==t 


Such  arrangements  are  well  adapted  to  the  organ.  The  second  chord 
is  the  essential  yth  to  F,  and  yet  the  section  begins  and  ends  in  B-JJat, 
But  the  student  should  understand  that  the  nature  of  this  discord  is 
almost  completely  changed  at  3,  and  that  these  changes  direct  it  intc 
a  different  channel.* 

The  following  illustrates  in  a  more  musical  manner  the  particulai 
application  of  this  chord : 

Jl'i/sitn  G.  Smith. 


Ex.  499. 


B&(\,     .     .      and  yearnings  sad      my  lone  heart  doth  en-dure. 


It  is  sometimes  desirable  to  raise  the  4th  and  6th  simultaneously  aftei 
having  lowered  the  real-base  : 


Ex.  500. 


"  The  discord  marked  I  is  a  fundamental  harmony;  the  one  marked  3  is  a  derived  bar- 
tnony. 


GOODRICH'S  ANALYTICAL  HARMONY. 


211 


The  student  should  be  familiar  with  the  various  methods  of  intro- 
ducing and  arranging  the  altered  6th  chords. 

Any  of  the  three  diminished  yth  chords  that  represent  the  related 
minor  keys  may  be  utilized  in  producing  the  discord  under  notice. 
This  is  partially  illustrated  by  the  chord  marked  II,  in  Ex.  498.)    In 
the  scale  of  D  these  diminished  chords  are  the  following : 


Ex.  501.  : 


In  their  present  appearance  they  represent  B-minor,  E-minor,  and 
/• ' '-minor.  Invert  each  one  twice,  lower  the  real-base  a  chromatic 
step,  and  the  result  will  be  three  augmented  6th  chords  of  the  3d 
species,  leading  to  G-major,  C-major,  and  D-major.  These  are  to  be 
worked  out  in  different  positions  as  directed.  *  *  * 

The  intervals  from  the  real-base  must  be :  a  major  3d ;  doubly- 
augmented  4th,  and  augmented  6th.  The  immediate  resolution  is  to 
a  major  chord  with  its  5th  in  the  base.  These  points  are  particularly 
important,  since  there  are  three  discords  that  sound  exactly  the  same, 
the  differences  being  in  their  notation  and  resolution.  These  are  here 
presented,  with  their  resolutions  indicated  by  quarter  notes : 


Ex.  502. 


•*••• 


The  first  belongs  naturally  to  A-flat ;  the  second  to  G -minor ;  the 
third  to  G-major.  The  discord  at  (a)  is  an  essential  yth.  The  other 
two  are  augmented  6th  chords  of  the  first  and  third  species.  The 
different  signatures  show  the  different  situations  in  which  these  dis- 
cords would  naturally  occur.  The  minor  yth  at  (a)  becomes  an  aug- 
mented 6th  at  (b) ;  and  the  normal  5th  at  (b)  appears  as  a  doubly- 
augmented  4th  at  (c).  Examples  (b)  and  (c)  should  be  worked  out 
to  their  final  tonics. 

The  examples  explained  and  illustrated  in  this  chapter  are  to 
be  transposed  (both  theoretically  and  practically)  into  several  other 
scales,  and  every  exercise  should  be  written  in  three  positions.  The 
different  modes  of  treatment  with  regard  to  the  endings  are  also  to 
be  used  in  different  examples. 


212  GOODKICH'S  ANAT_YTICATV  HARMONY. 

The  derivation,  notation,  and  resolution  of  the  various  augmented 
6th  chords  i,  2,  and  3  must  be  thoroughly  understood,  for  in  the  next 
chapter  attempts  will  be  made  to  use  them  in  the  harmonization  of 
themes,  as  they  are  employed  in  actual  composition. 


Chapter  XLVI. 


APPLICATION  OF  THE  VARIOUS  AUGMENTED 

SIXTH  CHORDS  IN  HARMONIZATION 

CONCLUDED. 

THE  three  principal  augmented  6th  chords  are  to  be  considered 
as  transition  chords,  even  when  their  resolution  is  indirect. 

No.  i  contains  a  major  3d,  normal  5th,  and  augmented  6th,  and 
resolves  to  a  minor  chord  in  its  second  inversion. 

No.  2  contains  a  major  3d,  augmented  4th,  and  augmented  6th. 
This  resolves  direct  to  a  major  chord  in  its  first  position. 

No.  3  consists  of  a  major  3d,  doubly-augmented  4th,  and  aug- 
mented 6th.  Its  immediate  resolution  is  to  the  second  inversion  of 
a  major  chord. 

The  final  resolution  of  No.  i  is  to  minor ;  No.  2  ends  in  major  or 
minor,  according  to  the  prevailing  tonality ;  No.  3  belongs  expressly 
to  the  major  mode.  The  immediate  resolution  of  i  and  3  determines 
the  final  resolution.  The  resolutions  are  all  intermediate,  and  must 
proceed  beyond  the  point  at  which  the  augmented  6th  chord  dis- 
appears. 

In  actual  practice  the  principal  difficulty  will  consist  in  properly 
introducing  these  altered  6th  chords.  First  of  all  decide  the  follow- 
ing points:  What  chord  do  you  wish  to  use  in  a  certain  place,  i,  2, 
or  3?  What  is  the  real-base?  What  other  chord  will  contain  one 
or  more  connecting  tones  ? 

Suppose   this  chord  was  required :     Ex.  503.  RgF  x;;^  It   is. 


GOODRICH'S  ANALYTICAL  HARMONY. 


213 


derived  primarily  from  the  second  inversion  of  a  secondary  yth  chord. 
The  root  will  be  A,  and  the  real-base  e.  So  much  being  ascertained, 
the  next  step  is  to  find  a  connecting  chord  with  this.  As  the  base 
is  to  be  <?,  the  preparatory  harmony  appears  naturally  upon  D  or  F. 
for  it  is  not  well  to  skip  to  an  inverted  base.  Arrangements  like  the 
following  may  serve  as  models  for  illustrating  this  process : 


or    2 


Ex.  504. 


The  first  preparatory  chord  marked  +  contains  one  connecting  note 
with  the  derivative  7th  chord,  and  all  the  parts  move  by  natural  de- 
grees. In  the  second  exercise  the  F  chord  furnishes  two  connecting 
notes.  Observe  also  the  melodic  progression  of  the  base.  (Transpose 
Ex.  504  into  D  and  B-flat?)  A  theme  for  harmonization  is  here  pre- 
sented, the  particular  object  being  to  introduce  the  augmented  6th 
chords  in  a  practical  and  musical  manner  : 

Ex.  505.  _k^. 


/m.  *    >    ?    •    • 

*  *  ;    i 

1 

1  ... 

1     L_l_^  

\      r 

^ 

{m^r  i  rr  r 

•f  i  M 

'        1 

q  

1  —^ 

1 

1. 

2. 

3. 

c&4-f-p  



J^^  1  

-&  

\          f 

^H 

—  ^       %               \ 

&  .i 

(2) 

The  figures  i,  2,  3  refer  to  the  different  species  of  augmented  6th 
chords.  The  last  two  ^'s  in  the  first  measure  signify  that  the  discord 
marked  i,  and  the  following  minor  triad  to  which  it  resolves,  both 
contain  c,  This  minor  triad  must  appear  with  its  5th  in  the  base, 
followed  by  the  dominant  and  tonic.  The  chord  at  2  resolves  direct 
to  a  major  chord  (dominant),  after  which  the  minor  yth  is  introduced 
— c  being  retained  in  the  middle  part.  3  indicates  an  augmented  6th 
chord  of  the  third  species,  resolving  to  the  second  inversion  of  a 
major  chord.  This  is  a  transition  to  G-major.  The  first  two  half 
notes  represent  two  quarter  notes  each  in  the  other  parts. 

Illustrations  and  explanations  in  the  previous  chapters  may  be 
referred  to. 


214 


GOODRICH'S  ANALYTICAL  HARMONY. 


After  completing  the  harmonization  in  four  parts  it  should  be 
written  in  two  other  positions  with  the  same  base.  The  mezzo-so- 
prano part  may  be  copied  as  a  theme,  but  the  original  contralto  part 
should  be  inverted  an  octave  higher  when  it  appears  as  melody. 

Afterwards  transpose  into  A-flat,  B-flat,  D,  and  E-flat. 

(Those  who  have  not  the  benefit  of  an  instructor  will  find  the 
solution  in  the  Key,  but  the  author  again  urges  students  not  to  con- 
sult this  part  of  the  work  except  when  necessary.) 

Another  theme  is  given  writh  the  same  general  object  in  view : 

Ex.  506. 


JLJa. 

r 

J  a 

i      '     i 

"      i     r 

r?Tr 

•  & 

0    9    &    \ 

i     *V 

1 

1 

i      i           i 

lill 

Tr 

1       1 

• 

1 

J 

1. 

2. 

3. 

• 

C""\* 

I             _ 

i 

J. 

f 

! 

\    3 

28 

—  .- 

f    # 

-^   rt 

..3 

»* 

_' 

* 

J 

2 

ty 

22 

1 

1     1 

BJI 

i-  ' 

j 

(2) 

Modulations  are  here  made  to  D-minor,  G-minor,  and  E-flat-major, 
by  means  of  the  augmented  6th  chords,  in  addition  to  the  passing 
modulations  through  the  essential  jth  chords.  The  resolutions  here 
are  the  same  as  in  the  theoretical  exercises.  The  only  difficulty  con- 
sists in  ascertaining  what  particular  key  is  intended  at  the  different 
points  indicated  by  the  figures.  These  latter  tell  the  species  of  6th 
chord  to  be  used,  and  as  only  the  related  keys  are  to  be  reached, 
students  who  have  learned  how  to  apply  mental  force  will  have  no 
trouble  in  completing  the  harmonizations  as  intended. 

During  the  fifth  and  sixth  measures  the  regular  formula  as  to 
intermediate  and  final  resolutions  is  slightly  varied.  An  avoided 
cadence  is  indicated  by  the  dash,  in  place  of  the  tonic  resolution  to 
E-flat-major.  Avoided  cadences  in  such  situations  are  always  proper 
and  usually  effective.  *  *  * 

The  altered  6th  chords  introduced  into  the  last  themes  have  be- 
come known  as  "  Italian,"  "  German,"  and  "  French  Sixths,"  though 
for  \vhat  reason  none  can  tell.  These  chords  can  not  be  nationalized, 
and  the  names  have,  therefore,  no  significance  whatever. 

The  augmented  6th  chord,  No.  2,  is  the  most  masculine  and  inci- 
sive of  all  the  transition  chords.  The  resolution  of  these  intervals, 


Ex.  507. 


GOODRICH'S  ANALYTICAL  HARMONY. 


215 


ascending  and  descending  by  half  steps,  produces  a  very  decided 
effect,  while  the  two  major  thirds  add  considerable  boldness  to  the 
transition.  The  augmented  4th  (d)  supplies  the  dissonant  quality 
(aside  from  the  sharpened  6th),  for  d  produces  a  discord  with  a-Jlat, 
as  well  as  with  c. 

Composers  frequently  omit  the  augmented  4th,  even  in  full  har- 
mony. In  such  instances  the-  3d  (from  the  real-base)  is  usually 
doubled,  and  then  resolved  differently  in  each  part,  thus; 


,/„, 


Ex.  508. 


The  upper  c  descends  to  b,  while  the  lower  c  ascends  to  d,  leaving 
the  G  chord  complete.  Besides,  it  is  usually  better  to  resolve  a  du- 
plicated tone  either  in  contrary  or  oblique  movement.  This  may  be 
done  whenever  the  augmented  4th  can  not  be  introduced  conven- 
iently. 

There  is  another  form  of  augmented  6th  chord,  consisting  of  an 
essential  discord  in  its  fourth  position  with  the  original  5th  sharp- 
ened : 

Ex.  509. 


But  this  is  generally  accompanied  with  the  fundamental  in  the  base, 
as  here : 


Ex.  510. 


The  resolution  of  the  essential  discord  is  not  materially  changed  on 
account  of  the  altered  interval,  and  as  the  chord  here  stands  it  is 
simply  a  dominant  ~th  with  augmented  5th.  Hence  it  belongs  more 
to  the  fundamental  harmonies  than  to  the  altered  6th  chords. 

Augmented  6th  chord,  Xo.  2,  is  sometimes  treated  in  the  same 


2l6 


GOODRICH'S  ANALYTICAL  HARMONY. 


way,  the  original  root  being  given  the  base.  When  this  occurs  the 
6th  chord  proper  resolves  as  usual,  while  the  independent  base  skips 
from  dominant  to  tonic : 


Ex.  511. 


-P-ir 


m 


This  is  a  somewhat  rugged,  forcible  harmonization,  and  its  use  i» 
rare.  It  also  brings  the  base  into  greater  prominence  on  account  of 
its  independent,  fundamental  character. 

The  augmented  6th  chords,  1,2,  and  3,  are,  however,  the  most 
important  and  characteristic.  With  the  derivation  and  resolution  of 
these  the  student  should  be  familiar  in  every  practicable  key. 

This  subject  will  be  concluded  with  a  few  quotations  from  stand- 
ard works,  intended  to  show  the  deviations  from,  rather  than  the 
conformations  to,  our  theoretical  formulas.  The  first  extract  is  from 
the  allegro  to  Mozart's  2d  G- minor  Symphony : 


Ex.  512. 


* 


:# 


7* 


S 


Mrj3ti»  -^. 

The  6th  chord,  No.  2,  does  not  here  resolve  directly  to  the  dominant 
harmony,  but  to  the  inverted  tonic  chord  first.  This  is  on  account 
of  the  melody  above,  g,  a,  b-flat.  Observe  the  chromatic  progression 
in  the  tenor  and  base  parts,  as  the  augmented  6th  chord  is  produced 
principally  by  this  contrary  movement.  The  next  extract  is  from 
the  "Jupiter"  Symphony  : 


v 


Ex.  513. 


No.  3. 


i^%4  4  | 


1 


3P 


GOODRICH'S  ANALYTICAL  HARMONY. 


217 


The  quotation  commences  at  the  end  of  a  transitional  section  in  D- 
flat,  and  this  shows  the  manner  of  returning  to  the  original  tonic,  C. 
The  second  chord  in  the  first  measure  is  a  passing  diminished  har- 
mony on  the  dominant,  a-flat.  By  changing  the  essential  yth  chord 
enharmonically  there  results  an  augmented  6th  chord,  No.  3,  resolv- 
ing naturally  to  C. 

In  the  andante  to  the  same  symphony  the  composer  has  resolved 
the  base  first,  and  then  the  other  parts  alternately,  thus : 


I  Mozart. 


Ex.  514. 


Observe  that  the  treble  parts  are  suspended  after  the  lower  resolu- 
tion, and  the  c  does  not  move  to  b  until  after  the  second  beat.  This 
may  be  done  with  any  four-toned  discord. 

The  next  illustration  is  still  more  exceptional.     It  is  from  the 
slow  movement  to  the  2d  G-minor  symphony  : 

Mozart. 


5*5- 


Observe  firstly  the  two  lower  staffs.  On  the  third  beat  an  augmented 
6th  chord  appears,  and  in  the  resolution  o -sharp  descends  to  g-natu- 
ral,  in  place  of  ascending  to  a.  But  the  sequence  embraces  a  chro- 
matic progression  similar  to  the  succession  of  essential  yth  chords. 
B-flat  appears  in  the  base  on  account  of  the  prevailing  tonality,  and 
this  is  what  produces  the  augmented  6th  chord  in  connection  with 
the  dand  g-shar.p  above.  Observe  the  two  ^'s  moving  contrarily, 


218 


GOODRICH'S  ANALYTICAL  HARMONY. 


and  especially  the  chromatic  progression  in  the  lower  treble  part 
The  violin  figure  above  is  a  variation  of  the  middle  parts. 

The  student  is  to  be  cautioned  against  writing  a  succession  of 
such  progressions  as  this  from  the  last  quotation : 


Ex.  516. 


Their  effect  is  the  same  as  two  minor  sevenths,  which  can  not,  as  a 
rule,  follow  each  other.  Several  such  dissonant  successions  would 
be  liable  to  offend  good  taste,  and  even  a  single  progression  of  these 
intervals  must  be  justified  by  some  such  design  as  the  one  quoted 
from  Mozart. 

Another  exceptional  resolution  of  an  augmented  6th  harmony  is 
here  cited  from  The  Dream  of  Jubal,  by  A.  C.  Mackenzie  • 


Ex.  517. 


There  is  nothing  unnatural  about  this,  and,  doubtless,  similar  in- 
stances exist.  By  including  c  in  the  tenor  a  connecting  tone  would 
result,  and  this  might  be  utilized. 

By  lowering  the  5th  of  No.  i  a  minor  2d  it  will  result  in  a  similar 
chord  of  the  2d  species,  and  ma}r  therefore  go  direct  to  the  dominant 
chord.  Such  an  instance  is  here  extracted  from  the  F-minor  con- 
certo by  Chopin : 


Ex.  518. 


The  change  from  i  to  2  at  +  may  be  considered  an  expediency,  an 
in  such  situations  a  most  useful  one.  Beethoven  frequently  change^ 
an  augmented  6th,  No.  i  into  No.  2,  when  he  desired  to  pass  direcii> 


GOODRICH 'S   ANALYTICAL    HARMONY. 

to  the  uninverted  major  chord,  and  to  avoid  parallel  fifths.  In  these 
instances  the  minor  chord  in  its  second  inversion  does  not  appear, 
but  the  final  resolution  is  the  same. 

In  a  descending  sequence  of  dominant  yth  chords  an  augmented 
6th  chord  of  the  first  or  third  species  is  sometimes  substituted  at  the 
close,  as  a  means  of  establishing  more  directly  a  particular  key.  This 
is  partially  explained  by  E::3.  .^02  and  513.  The  student  should  write 
such  exercises,  using  augmented  6th  chord,  i  for  the  minor,  and  3 
tor  the  major,  cadence. 

Examples  of  this  will  be  included  in  the  Key. 


22O 


GOODRICH'S  ANALYTICAL  HARMONY. 


PART  XI. 


Chapter  XLVII. 


HARMONIC  PROGRESSIONS  IN  GENERAL. 
THEIR  ESTHETIC  EFFECT. 


HERETOFORE  chord  succession  has  been  considered  principal! 
in  its  fundamental  order  and  relation.  Excepting  in  the  t\v 
chapters  relating  to  harmonic  cadences  the  author  has  refrained  fror 
presenting  chord  progression  in  its  purely  euphonious  aspect.  Th 
harmonic  cadences  will  serve  as  a  basis  for  this  study,  which,  in  it 
general  features,  is  a  rather  superficial  one. 

The  relations  of  tonic  and  dominant  are  so  intimate  that  the 
admit  of  unlimited  alternate  repetition,  after  the  manner  of  an  at 
thentic  cadence.  The  following  stretto  illustrates  this; 


Allegro. 


Ex.  519. 


-*   •   r 


.*• 


f  <>«»». 


=g— s*-g 

•»••*••*• 


^'53= 


-Jr       -f- 


No  other  two  harmonies  have  been  used  so  frequently  as  these,  not 
withstanding  their  affirmative  character. 

The  subdominant  is  next  in  order.  On  account  of  its  connectioi 
with  the  tonic  harmony  the  subdominant  maintains  the  tonic  impres 
sion  more  nearly  than  does  any  other  chord : 


GOODRICH  S    ANALYTICAL    HARMONY. 


Ex.  520. 


221 

Schiilz. 


The  tonic  remains  in  the  base  throughout  this  section  and  the  changes 
to  subdominant  in  measures  3  and  7  represent  the  least  disturbance 
of  tonic  impression ;  for  the  theme  might  have  been  accompanied 
exclusively  with  the  tonic  chord  without  materially  altering  the  effect. 
In  the  cadence-forms  these  two  chords  were  considered  in  their  well- 
known  capacity  as  constituting  an  after  cadence.  They  are  not  to  be 
so  regarded  here,  for  the  last  example  is  an  initial  section,  not  a  final 
close.  The  subdominant  harmony,  preceding  that  of  the  tonic,  is 
extremely  mild  and  does  not  possess  that  positive  quality  that  is  char- 
acteristic of  dominant  and  tonic.  They  are  in  fact  almost  opposite 
in  their  effects,  for  one  is  progressive,  the  other  is  non-progressive. 
Compare  Exs.  520  and  521  for  this  purpose: 

Ex.  521. 


£ 


(TO  * 

1       .  .._• 

1  —  i  —  i 

—  I 

SF  

*.*-*.          +.+.*.           +.+.+. 

R:>  
5^5—  » 

S     S     S 

Z       Z       Z               0r     *r     ~ 

8^1 

—  r     1  —  I  — 

-r-                                 —    '  T 

This  is  very  simple  pabulum  from  Czerny,  but  it  serves  the  present 
purpose.  Observe  how  much  more  decided  are  the  changes  of  har- 
mony in  the  last  example. 

By  combining  these  three  fundamental  harmonies  we  have  the 
characteristic  effects  noted,  and  at  the  same  time  the  most  pleasing 
and  easily  comprehended  means  of  harmonization.  And  it  must  be 
said  that  they  have  been  used  innumerable  times  for  mere  ear-tingling 
purposes.  Haydn  employed  them  almost  constantly,  Mozart  less  so, 


222 


GOODRICH'S  ANALYTICAL  HARMONY. 


and  the  popular  opera  composers  have  worn  them  threadbare.  I 
his  mature  works  Beethoven  made  very  little  use  of  the  perfe< 
cadence-form.  Schumann  considered  it  a  symptom  of  philistinism 
Chopin  was  rich  in  harmonic  invention ;  Berlioz  and  Wagner  d< 
spised  all  such  devices,  and  at  the  present  time  these  cadence-ha 
monies  have  fortunately  passed  into  a  state  of  inusitation.  The 
chief  value  to  the  young  composer  is  to  furnish  a  simple  and  euplu 
nious  basis  upon  which  to  erect  a  broader  and  more  diversified  ha: 
monization.  Even  in  an  elementary  chapter  acquaintance  was  mad 
with  thirty  chord  progressions  in  a  single  major  scale,  and  all  thes 
are  of  practical  utility.  Since  then  hundreds  of  chord  succession 
have  been  herein  illustrated,  and  it  only  remains  to  note  the  rels 
tionship  of  certain  harmonizations  and  observe  the  effect  of  differer 
orders  of  chord  movement.  A  quotation  from  the  allegro  to  Bee 
hoven's  Adelaide  is  presented  : 
Ex.  522. 


ft  -  (t 


-1 — I— + 


s 


-x— •- 


-f- — f — I*- 


fcFS=M: 


•W  ^F~ 

J=t 


The  tonic  chord  occurs  but  once ;  the  subdominant  twice  (the  laj 
time  inverted)  ;  and  the  following  chords  once  each  :  C-minor,  F-»u 
jor,  D-minor,  G-minor^  and  finally  the  dominant  yth  harmony.  Thi 
scheme  introduces  all  the  concords  in  B-flat-major  (besides  the  esser 
tial  yth  chord),  and  these  occur  principally  in  a  dominant  relatior 
thus:  B-flat  to  E-flat ;  Ctof;  and  D  to  G.  These  form  sequence; 
There  is  also  an  alternate  third  relation  :  E-flat  to  C ;  F  to  D ;  an 
G  to  E-flat.  In  the  former  instance  there  is  one  connecting  tone  ;  i 
the  latter  there  are  two.  This  proves  that  a  perfectly  natural  an 
euphonious  series  of  harmonies  may  consist  of  something  more  tha 
tonic,  subdominant  and  dominant,  and  without  even  a  temporar 
transition. 


GOODRICH'S  ANALYTICAL  HARMONY. 


223 


The  next  quotation  is  of  a  different  nature.     It  is  the  second 
theme  in  Schubert's  Unfinished  Symphony  : 


Ex.  523. 


-*-— 


Ef 


£=»: 


-»-r* 


1 


I 


N  N 

I         rfrJ*     ->-  -««- 


S 


Onl}'  a  limited  number  of  harmonies  are  here  employed,  for  the  grace 
and  simplicity  of  the  melody  require  very  little  elaboration.  The 
syncopated  middle  parts,  the  chord  figure  in  the  base,  and  the  modu- 
lations to  and  from  A-minor,  are  to  be  noticed. 

Melodic  passages  that  remain  very  nearly  upon  a  monotone  re- 
quire more  harmonic  elaboration  than  do  such  themes  as  the  one  last 
quoted.  The  very  nature  of  the  case  makes  this  necessary,  for  if 
there  is  no  melodic  progression  in  the  upper  part,  there  must  be  in 
the  harmony.  In  vocal  music  this  is  of  frequent  occurrence,  espe- 
cially where  the  solo  part  is  of  a  declamatory  nature.  Such  an  in- 
stance is  the  following  from  Mililotti : 


THE    POOR    MARINER. 


Ex.  524. 


Aub                p__C_3 

1            ^ 

P         C     W      •  iT               -       ^ 

ffTr       «      «      «N 

i 

— 

V             *            *S                                   41          *       * 

* 

«/ 
('old-ly     the  dav     is    end-ing,      Niujht  fmm  the  licav'ns  descciid-ing; 

1        y       -'    "                             ^ 

jjf_  t,  — 

—  |  —  |  

F?V>^             i 

<^~                     --*                /^L 

"  V                                               + 

**:                                   w3?~« 

^     &            i  '      % 
"9"               w         "f 

Z               '3Z.  " 
•                   "ft*                              •*                 ~2^» 

.j                     ,    i                 j 

9m  i^   s~              i^ 

-5. 

"*!                          rtM 

fj  K        -^                      y&i 

£24 


GOODRICH'S  ANALYTICAL  HARMONY. 


JV      £       JS K^P^-^-""^ 

-V*--V+^—0  •  -L- 


Hoars  - !/    the   bil-lows  moaning,     Sounds  in    the  fate-ful  gloaming; 


Here  are  seven  different  harmonies  accompanying  the  g  and  b-flat  oC 
the  voice-part,  which  is  almost  non-melodious.  The  design  is  well 
conceived  and  worthy  of  careful  study.  Somewhat  similar  instances 
occur  in  the  Farewell  Song,  by  Schubert,  and  in  the  Palms,  by  Faure. 
These  are  so  nearly  alike  that  an  epitome  of  the  latter  only  is  given : 


E.X.  525. 


53=    ^^==^z^=3^=: 

? — * — -^  f    §»~^~» —          — b*—  - 


-*-=-- 


3 


etc. 


This  scheme  is  easily  explained.  The  3d  of  a  dominant  yth  chord 
descends  by  half  steps  to  the  root-tone,  and  the  root  ascends  chro- 
matically to  the  3d.  Meanwhile  the  5th  and  yth  remain  stationary, 
as  parts  of  the  intervening  harmonies. 

Transpose  and  re-arrange  this  example,  omitting  the  upper  part, 

A  somewhat  similar  instance  is  extracted  from  the  opera  Puritan  i  a  • 

EX.  526.  R.  S.   Kclley. 


-**'r—  A  *£    -  '  ".•'*— 

-* — f:  tfrrw^ri    ~f^!^rr-r 

P  U  U     '      '  V  V     \      \ 


•jt 

-• 


I  I 

S  n 


1 


—¥- 


:; 


-•— »• 


-«•:*• 


GOODRICH'S  ANALYTICAL,  HARMONY. 


225 


Owing  to  the  monotonous  character  of  the  melody  the  harmonic  parts 
are  considerably  varied.  Observe  the  base  progression  from  a  down 
to  b,  and  how  the  cadences  are  avoided  until  the  close. 

In  the  first  part  of  Lohengrin,  after  the  king's  solo,  there  are  some 
progressions  that  have  been  classed  as  "  inharmonious  "  by  certain 
theorists.  For  instance,  F  to  E-flat ;  F  to  C-mi?ior ;  G  to  F-minor, 
and  so  on.  This  proves  what  has  frequently  been  stated  in  this  vol- 
ume, that  any  chord  progression  is  correct  if  employed  effectively. 
Besides,  have  we  not  heard  too  much  of  subdominant  and  dominant  ? 

The  re-arrangement  or  inversion  of  certain  harmonic  and  melodic 
designs  may  be  included  here,  especially  where  such  an  arrangement 
serves  to  carry  out  the  same  idea  : 


Ex.  5270. 


The  motive  at  (a)  is  repeated  at  (b)  in  the  base  part,  while  the  melody 
above  is  continued.  This  is  not  to  be  confused  with  the  inversion 
of  separate  chords,  as  here : 


Ex.  527^. 


JECven  these  may  prove  useful  in  harmonization,  though  no  harmo*  ic 
progression  takes  place  in  the  different  measures  considered  st;  -a- 
rately.  Beethoven  produced  many  novel  effects  by  means  of  inver- 
sion, as  with  the  26.  theme  in  the  finale  to  his  Op.  27,  No.  2. 


226 


GOODRICH'S  ANALYTICAL  HARMONY. 


In  the  following  quotation  from  Wagner  a  derived  harmony  as- 
sumes a  fundamental  character  on  account  of  the  added  base  below  •. 


Ex.  528. 


The  passing  diminished  chord  above  is  made  much  more  incisive  by 
the  fundamental  progression  in  the  base — dominant  to  tonic. 

Since  the  most  natural  order  of  successive  harmonies  has  fre- 
quently been  reversed  it  would  be  useless  to  prescribe  certain  for- 
mulas. The  present  object  is  to  call  attention  to  various  methods 
and  means  of  harmonization,  and  point  out  their  peculiar  effects  and 
applications.  Fundamental  progressions  by  seconds  are,  on  account 
of  their  want  of  connection,  inclined  to  be  abrupt  and  rugged. 

Progressions  by  thirds  (fundamentally)  are  most  intimately  con- 
nected and  related,  and  produce  an  opposite  effect  to  that  of  the 
disconnected  progressions.* 

The  ascending  fourths  are  sufficiently  connected,  and  have  a  dom- 
inant relation  that  represents  something  of  incitement  and  onward 
progress. 

The  progressions  by  fifths  ascending  (or  fourths  descending)  are 
also  connected  by  a  tone  in  common ;  but  they  have  a  retrogressive 
tendency. 

Chromatic  movements  have  been  described  in  a  general  way,  but 
their  changing  colors  are  so  kaleidoscopic  that  no  one  could  hope  to 
explain  them  in  detail.  Since  the  advent  of  Chopin  and  Schumann 
the  tendency  has  been  to  indulge  very  freely  in  chromatic  transition 
and  elaboration.  The  postlude  to  one  of  Schumann's  songs  is  quoted 
as  a  simple  illustration  : 

Ex.529.  ~    "^ 


•-This  important  relation  by  the  sd  is  partially  explained  by  the  natural  series  of  har- 
mcnic  tones.     But  it  is  not  necessary  to  discuss  this  here. 


GOODRICH  S    ANALYTICAL    HARMONY. 


227 


—    *-.*- 


Almost  every  chromatic  passing  tone  is  here  employed.     The  effect 
is  appropriate  -and  charming. 

We  have  become  so  accustomed  during  recent  years  to  almost 
incessant  transition  that  a  passage  like  the -following  scarce  disturbs 
the  key-impression : 

J.  Low,r.Op.  485. 


This  begins  and  ends  in  C,  but  it  includes  a  phrase  in  A-flat  and  one 
in  F-minor. 

In  connection  with  this  subject  students  are  recommended  to 
study  as  many  as  possible  of  the  following  scores  :  Schumann,  Ops.  21 
and  26  ;  Ball-Scenes,  Op.  26,  Nicode ;  Moszkowski,  Spanish  Dances, 
Op.  12,  especially  the  No.  3,  from  the  rogth  measure  to  the  end  of 
the  movement ;  Mackenzie,  The  Song  of  the  Sickle,  from  Dream  of 
Jubal,  first  part  in  A-minor ;  Wagner,  Evening  Star,  Romance;  I. 
Low,  Paul  and  Virginia,  Op.  485 ;  any  of  Chopin's  piano  works ; 


228  GOODRICH'S  ANALYTICAL  HARMONY. 

Grieg's  Overture,  In  Autumn  ;  Franz,  The  Dark  Eye,  Op.  9,  No.  2. 
This  exquisite  song,  also  known  as  Request,  is  one  of  the  best  illus- 
trations of  rich  harmonization  to  be  found  even  in  modern  music. 
It  should  be  transposed  a  minor  2d  lower,  if  that  would  more  plainly 
show  its  peculiar  and  masterly  harmonic  structure.  Either  the 
original  song,  or  Mr.  Clarence  Eddy's  organ  arrangement,  should  be 
consulted.  The  latter  may  be  found  in  Vol.  II  of  The  Church  and 
Concert  Organist. 


Chapter  XLVIII. 


FIGURED  BASES.     A  CURSORY  VIEW. 

WHEN  the  author  of  this  volume  first  began  to  formulate  a 
system  of  Harmony  [in  the  year  1866]  he  discarded  the  then 
prevailing  "  Thorough  Base "  methods,  because  he  was  convinced 
that  figured  bases  had  served  their  purpose,  and  could  not  be  made 
sufficiently  precise  or  comprehensive  to  support  a  practical  theory 
of  composition.  From  that  date  unto  the  present  he  has  totally 
ignored  the  "  thorough  base  "  formulas ;  but  so  many  books  have 
been  published  on  this  plan,  and  so  many  old  scores  contain  figured 
bases,  that  he  has  concluded  to  give  a  brief  explanation  of  the  sub- 
ject, independently  of  his  own  system. 

Thorough  base  represents  a  method  of  musical  stenography, 
wherein  a  single  base  part,  accompanied  by  figures,  indicates  the  full 
harmony.  To  ascertain  the  corresponding  note  of  any  figure  it  is 
only  necessary  to  count  up  from  the  base.  The  computation  is  al- 
ways made  from  the  actual  base,  this  being  counted  i.  The  intervals 
of  a  triad  from  the  root  are  a  third  and  fifth.  Therefore  the  base  note 
C,  marked  £  signifies  that  the  chord  of  C  is  to  be  played.  The  inver- 
sions are  numbered  according  to  the  actual  distance  of  the  intervals 
from  any  real-base.  If  e  in  the  base  is  to  represent  the  C  chord,  it 
is  figured  \ .  3  represents  g  and  6  represents  c.  Together  these  pro- 
duce the  full  chord,  e,  g,  c,  in  the  first  inversion.  In  the  second 
inversion  g  is  the  real-base.  The  remaining  tones  of  the  chord  of  C 
are  situated  a  fourth  and  a  sixth  above.  Therefore  the  g  would  be 


GOODRICH'S  ANALYTICAL  HARMONY. 


229 


figured  I  to  indicate  the  second  inversion  of  the  C  chord.  The  first 
inversion  is  usually  figured  6,  the  3  being  omitted.  When  no  figures 
are  included  the  base  is  a  root. 

Here  are  a  few  simple  illustrations : 


Ex.  531. 


f 


-& 


(a)  and  (b)  show  the  manner  in  which  the  harmony  would  be  added 
to  the  base  part  in  three  and  in  four  parts.  At  (c)  the  —  shows  that 
the  C  chord  is  sustained  above,  while  the  base  passes  through  the 
various  notes  of  the  same  chord.  The  figuring  of  the  dominant  yth 
chord  and  its  inversions  is  as  follows : 


Ex.  532. 


7  signifies  that  a  chord  of  the  yth  is  required,  the  base  being  the  root. 
|  indicates  the  first  inversion — 3  being  presupposed;  |  shows  the 
second  inversion — 6  being  omitted,  or  presupposed.  The  last  inver- 
sion calls  for  *  (g,  b,  d\  but  this  has  been  abbreviated  as  shown  in 
the  last  example.  Any  position  may  be  written  above,  provided  the 
chord  is  generally  correct.  For  instance,  the  chord  of  the  third  and 
fourth  may  be  taken  in  any  of  the  following  arrangements : 


Ex.  533 


The  figures  \  simply  indicate  the  second  inversion  of  a  yth  chord, 
and  so  all  these  arrangements  are  correct.  This  is  one  of  the  prin- 
cipal objections  to  the  system  ;  for  a  performer  can  not  tell  with  any 
degree  of  certitude  what  particular  design  the  composer  had  in  mind 
when  he  wrote  the  base  part. 


230  GOODRICH'S  ANALYTICAL  HARMONY. 

All  7th  chords,  whether  principal  or  secondary,  are  figured  in  the 
same  manner.  Transitions  are  indicated  by  placing  before  a  certain 
figure  whatever  chromatic  sign  may  be  necessary  for  the  correspond- 
ing note  above.  Thus  the  inversions  of  the  principal  diminished  yth 
chord  in  a  minor  key  would  be  figured  : 


Ex.  534. 


m 


Aside  from  the  sharp  the  figuring  is  the  same  as  that  employed  for  a 
dominant  or  secondary  yth  chord.     As  an  epitome  of  what  has  been 
thus  far  explained  a  short  exercise  in  figured  bases  is  presented  to 
be  Avorked  out  by  the  student : 
Ex.  535. 


0  -  IS 


J36  4  266  6  —  7- 

53  44- 

The  first  chord  is  to  have  its  fifth  uppermost.  After  that  it  is  only 
necessary  to  follow  the  general  principles  of  chord  progression  and 
the  rules  of  resolution.  The  second  base  note,  without  figures  is  a 
root ;  therefore  the  E-minor  triad  is  intended.  The  formula,  4-7, 
is  frequently  employed  in  final  cadences. 

No  further  explanation  is  necessary.     *     *     * 

According  to  the  principle  of  figuring  just  explained  various  har- 
monic combinations  are  indicated  from  the  base  part.  In  order  to 
carry  out  the  system  so  as  to  meet  all  requirements  the  figures  become 
almost  innumerable  and  frequently  confusing.  The  author  admits 
that  considerable  practice  of  this  kind  is  beneficial  to  the  student. 
But  the  fact  remains  that  information  thus  acquired  is  both  mechan- 
ical and  superficial.  One  never  knows  why  he  makes  a  certain  pro- 
gression according  to  the  "  thorough  base  "  formulas,  and  all  the  work 
becomes  artificial  and  uninspiring.  It  is,  in  fact,  beginning  at  the 
wrong  end,  for  all  music  is  founded  upon  melody,  and  the  harmony 
is  its  accompaniment.*  Composers  no  longer  employ  this  lazy-man's 
expediency,  and  it  certainly  ought  to  be  classed  among  things  that 
are  obsolete. 

Herewith  is  presented  a  solution  in  piano  score  of  the  "  thorough 
base  "  Ex.  535  for  comparison  : 

"Even  in  purely  harmonic  combinations  there  must  be  melodic  progression  in  some  ot 
the  voice-parts.  The  author  has  demonstrated  this  in  a  previous  work. 


GOODRICH'S  ANALYTICAL  HARMONY. 


231 


Ex.  536. 


Considerable  of  this  is,  of  course,  optional.     For  instance,  the  third 
measure  might  have  been  arranged  in  this  manner : 


Ex.  537. 


As  the  object  was  to  give  merely  the  principal  features  of  this  old 
method,  further  examples  need  not  be  presented.  Viadana  and  Cata- 
lano  first  employed  the  figured  base  method  about  1598.  It  was 
called  basso  continue. 


Chapter  XLIX. 


THE  NATURAL  AND  MELODIC  MINOR  SCALES. 
THEIR  HARMONIES. 

THE  NATURAL  MINOR. 

HISTORICAL  research  has  discovered  a  great  variety  of  scales 
that  have  been  used  at  different  times  during  the  past  fifteen 
centuries.  Nearly  all  of  these  (including  the  ecclesiastical  modes) 
have  been  discarded,  with  exception  of  the  Ionian,  beginning  on  C. 
This  is  now  called  the  Normal  Major  Scale.  Modern  composers  have 
undoubtedly  considered  the  minor  as  a  derived  scale,  for  it  has  since 
the  time  of  Bach  been  written  in  five  different  ways.  Most  of  these 
are  variations  of  the  old  ^olian  mode.  This  primitive  scale  is  de- 


232 


GOODRICH'S  ANALYTICAL  HARMONY. 


serving  of  more  attention  than  it  has  received,  and  the  author  believes 
it  will  be  employed  more  freely  in  the  future.  It  consists  of  a  natural 
series  of  seven  notes,  and  is  the  same  ascending  and  descending : 


EX.  538-  t*===z: 


1       a      3      4      5      6      7 


6       5       4       3 


The  two  small  steps  are  here  more  equally  distributed  than  they  are 
in  the  melodic  minor.  The  group  i,  2,  3  has  its  counterpart  in  4, 
5,  6 ;  the  major  element  being  represented  by  3,  4,  5,  with  its  corre- 
sponding sequence,  6,  7,  8.  Likewise,  a  counterpart  of  2,  3, 4  is  found 
in  5,6,7: 


Ex.539.fcjfc= 


Two  corresponding  tetrachords  may  also  be  formed  by  considering 
4  as  the  ending  of  one  and  the  beginning  of  another : 


Ex.  540. 


I 


These  characteristic  scale  features  are  usually  overlooked,  but  they 
were  considered  of  vital  importance  by  our  musical  forefathers. 

The  harmonization  of  this  scale  forms  a  prominent  part  in  its 
consideration.  Those  who  hear  an  old  melody  clothed  in  the  garb 
of  modern  harmony  have  no  better  notion  of  the  original  effect  than 
can  be  obtained  from  listening  to  one  of  Scarlatti's  harpsichord  sona- 
tas performed  in  the  maudlin,  tempo  rubato  style  of  the  present  day  ! 
But  we  are  so  accustomed  to  certain  harmonic  progressions  that  what- 
ever is  at  variance  with  the  prescribed  order  is  by  many  thought  to 
be  incorrect.  In  fact,  the  author  has  been  gravely  informed  by  a  con- 
vention of  music-teachers  that  such  progressions  as  these  are  wrong  J 


Ex.  541. 


It  is  no  fault  of  these  progressions  that  they  sound  bad  to  certain 
persons. 


GOODRICHVS   ANALYTICAL    HARMONY. 


233 


Though  the  natural  minor  scale  contains  a  minor  yth  from  the 
lower  tonic,  there  is  no  good  reason  why  the  dominant  major  chord 
may  not  be  used  in  the  lower  half  of  the  scale,  thus : 


Ex.  542. 


In  the  upper  tetrachord  the  natural  subtonic  is  used ;  in  the  lower 
the  modern  leading-note  is  included.  However,  instances  are  not 
wanting  in  which  the  subtonic  is  substituted  for  the  leading-note  at 
the  close.  This  is  according  to  the  natural  character  of  the  scale, 
whereas  the  regular  leading-note  is  a  foreign  element.  Reference  is 
made  to  the  final,  as  well  as  to  the  intermediate  cadence,  in  which 
the  dominant  minor  chord  is  employed.  Such  an  instance  is  given : 


Ex.  543. 


-« s- 

-7V      -*• 


:g r~— 

— id— H 


15: 


•^     i 


This  ambiguous  cadence  is  mild  and  plaintive,  and  ought,  therefore, 
to  serve  a  very  distinctive  purpose,  especially  where  decided  char- 
acter is  not  required.    A  similar  instance  from  Bizet's  La  Bohemienne 
is  quoted : 
Ex.  544. 


This  is  the  initial  period,  and  occurs  twice  with  the  subtonic  accom- 
panied by  the  E-minor  chord.  Rubinstein,  Rimsky-Korsakow,  Tschai- 
kowski,  Saint-Saens,  Dvorak  and  Grieg  have  produced  some  of  their 
most  characteristic  effects  by  means  of  these  quaint  harmonizations. 
The  Danse  Macabre  is  a  notable  instance,  for  no  leading-note  appears 
during  the  greater  part  of  the  work. 


234 


GOODRICH'S  ANALYTIC  AT.  HARMONY. 


It  should  be  added  to  what  has  been  said  about  Exs.  470,  543  and 
544,  that  the  fundamentals  remain  the  same,  thus  preserving  the 
dominant  relation  in  the  cadence.  On  this  account  they  are  more 
satisfactory  than  the  following  close  : 


Ex.  545- 


This  is  a  mere  progression  ;  as  a  cadence,  the  relative  major  sounds 
incongruous.     (See  Ambiguous  Cadence,  Chapter  XLJI.) 


THE  MELODIC  MINOR. 

This  is  a  modern  scale,  and  one  that  is  little  understood.  It  is 
derived  from  the  harmonic  form,  which  contains  a  minor  6th  and  a 
major  yth.  In  order  to  obviate  the  effect  of  an  augmented  2d  from 
6  to  7,  composers  frequently  raise  the  6th,  thus  making  the  last  of  the 
ascending  scale  exactly  like  that  of  the  major : 


546. 


m 


As  a  characteristic  series  of  sounds  this  is  inferior  to  the  other  forms, 
for  it  is  too  much  like  the  scale  of  A-major,  and  too  little  like  that  of 
A-minor. 

The  melodic  form  is  chiefly  valuable  in  a  rapid  ascending  ca- 
dence. The  following  extract  from  Mozart  shows  its  principal  ad- 
vantage ; 


Ex.  547. 


The  fact  that  Mozart  used  this  form  is  a  sufficient  reason  for  its  exist' 
ence,  but  the  manner  in  which  he  employed  it  must  be  borne  in 


GOODRICH'S  ANALYTICAL  HARMONY. 


235 


The  sharpened  6th  appears  as  a  mere  passing-note,  and  does 
not  receive  separate  harmonic  treatment.  This  has  led  to  the  erro- 
neous opinion  (asserted  as  a  fact  in  certain  thorough-base  books)  that 
this  scale  can  not  be  harmonized !  Considering  the  fact  that  even 
the  chromatic  scale  can  be  harmonized  in  various  ways  it  is  useless 
to  occupy  much  space  in  controverting  such  queer  notions.  It  would, 
indeed,  be  unfortunate  if  no  suitable  harmonies  could  be  found  for  a 
scale  so  frequently  employed  as  is  the  melodic  minor. 
A  few  examples  are  here  given : 

Ex.  548. 


h — I y^r^i ug-|>-rgT'fe/g&g i  ^  iHu.   *>£   gi^iKo   &g    T 

,-t 


ifT     LU 


-»•_•*. 


i'| 

rl  I 


Observe  that  the  melodic  cadence  is  the  same  in  each  phrase.    These 
are  correct  and  serviceable. 

In  the  descending  form  of  the  melodic  minor  scale  the  flattened 
7th  (subtonic)  is  sometimes  used  as  a  passing-note  to  the  minor  6th : 


Ex.  549- 


This  is  done  for  a  melodic  purpose. 

The  minor  jth  may  also  appear  as  an  appoggiatura  to  the  note 
below : 


-^       J-vJ 

-e—?-*-^\ 

=T     r       I 


Or  the  minor  yth  may  form  part  of  a  transition  chord  in  a  transient 
modulation  to  the  subdominant : 


*  See  also  Kuhlau  Sonatina,  Op.  55,  No.  3 ;   last  movement,  measures  21,  22,  23,  24,  of  the 
second  theme. 


GOODRICH'S  ANALYTICAL  HARMONY. 


Ex.551. 


In  the  second  measure  the  minor  3d  is  restored,  and  in  the  cadence 
the  leading-note  appears ;  so  the  example,  as  a  whole,  gives  a  very  fair 
representation  of  the  descending  scale  harmonized. 

The  concluding  example  illustrates  the  ascending  and  descending 
forms  of  this  scale,  following  each  other  immediately  : 


Ex.  552. 


The  sharpened  6th  is  used,  because  it  passes  more  naturally  to  the 
leading-note.  In  descending,  the  e-flat  is  restored,  because  it  is  part 
of  the  subdominant  harmony,  and  f-natural  leads  to  e-flat  more  me- 
lodically  than  would  f-sharp.  The  scale  is,  therefore,  an  expediency, 
both  ascending  and  descending,  and  aside  from  these  conditions  (as 
they  appear  in  the  last  example)  it  is  inferior  to  the  harmonic  form. 
If  a  composer  chooses  to  employ  the  melodic  form  in  certain  situa- 
tions, that  is  a  matter  of  esthetics  suggested  by  the  nature  of  his 
melody.  But  this  does  not  justify  the  assertion  that  the  harmonic 
form  is  non-melodious.* 

*  If  any  further  testimony  be  required,  let  the  spirit  of  Mozart  answer  through  his  Requi- 
em, or  the  ad  G-minor  Symphony. 


GOODRICH  S   ANALYTICAL    HARMONY. 


237 


Chapter  L. 


PRINCIPAL  AND  SECONDARY  NINTH  CHORDS. 

PRINCIPAL  NINTH  CHORDS 


most  important  five-toned  chords  are  the  dominant  gth  in 
major  and  the  dominant  9th  in  minor.    They  consist  of  a  major 
or  a  minor  3d  added  to  a  dominant  yth  chord,  thus 


Major  9th.    Minor  9th. 


The  root,  3d,  5th  and  yth  are  identical.  The  large  gth  is  perfectly 
natural  to  the  scale  of  G-major,  and  the  small  gth  is  equally  natural 
to  that  of  G-minor. 

All  these  combinations  are  double  dissonances,  though  they  do 
not  require  two  resolutions  when  they  are  dominant  9th  chords.  On 
account  of  its  dissonant  nature  the  gth  chord  should  be  prepared. 
Herewith  are  examples  showing  the  preparation  and  resolution  of  a 
major  gth  chord : 


Ex.  554. 


y 


-&-± — p^g — fg    1°    h^ ;    r 


iE 


e> 


i^ 


At  (a)  the  5th  is  omitted  to  prevent  parallel  fifths.  Besides,  the  3d 
is  more  essential  than  the  5th.  At  (b)  the  5th  and  gth  are  contained 
in  the  antecedent  yth  chord,  and,  therefore,  retained.  Otherwise  it 
might  not  be  well  to  omit  the  yth.  The  gth  chord  at  (c)  is  the  result 
of  a  melodic  progression  in.  the  soprano  and  contralto  parts.  As  a 
general  rule  the  9th  sounds  best  at  the  top.  The  reason  for  this 
is  that  the  combination  under  notice  contaias  a  double  dissonance, 


238 


GOODRICH'S  ANALYTICAL  HARMONY. 


Ex.  555. 


and  these  tones  sound  incongruous  and  con- 


fused it  they  come  too  near  together. 

The  tendency  of  the  gth  is  to  resolve  down  a  2d  to  the  5th  of  the 
concord  (or  to  the  root  of  the  dominant  7th  chord),  and  the  student 
is  not  advised  to  seek  an  exception  to  this.  The  remaining  notes 
are  treated  as  though  the  9th  did  not  appear,  for  this  combination  is 
merely  an  essential  yth  chord,with  a  major  3d  superimposed  upon  it. 
(See  Ex.  874.) 

In  five-part  harmony  the  full  Qth  chord  may  be  used,  but  this  is 
not  essential.  In  four-part  harmony  omit  the  5th,  yth,  or  3d. 

In  a  minor  scale  the  gth  added  to  the  dominant  7th  chord  will  be 
minor.  It  is  slightly  harsher  than  the  major  gth,  but  the  preparation 
and  resolution  are  governed  by  the  same  principles.  The  three  illus- 
trations in  Ex.  554  may  therefore  be  transferred  to  the  opposite 
mode: 


Ex.  556. 


m^ 


3: 


These  are  correct  and  effective. 

As  they  do  not  naturally  admit  of  re-arrangement,  it  will  be  suffi- 
cient to  transpose  them  into  several  other  scales.  *  *  * 

Ninth  chords  are  rather  difficult  of  management  when  inverted, 
because  the  ear  does  not  readily  recognize  their  tonal  structure  when 
the  root  is  displaced.*  The  following  arrangements  have,  howevei, 
been  employed : 


Ex.  557- 


"••Schumann,  in  his  B  flat  Symphony,  used  a  major  9th  chord  in  its  third  inversion.    See 
measures  72  and  73,  first  allegro. 


GOODRICH'S  ANALYTICAL  HARMONY. 


239 


The  5th  of  the  9th  chord  might  have  been  omitted  from  examples 
(a)  and  (c),  but  at  (b)  the  five  parts  are  indispensable. 

SECONDARY   NINTH   CHORDS. 

These  do  not  perform  any  direct  act  of  modulation  or  resolution, 
but  merely  form  parts  of  a  chord  progression,  thus: 


Ex.  558. 


The  first  two  are  secondary,  the  last  two  are  principal  gib.  chords, 
though  they  are  here  all  treated  as  preparatory  discords. 

In  the  Wedding  March  (2d  period)  Mendelssohn  used  a  secondary 
9th  chord  on  the  subdominant : 


Ex.  559. 


etc. 


g^=1=f^: 

-i *-^.-3: 


~at~ 

This  is  a  very  harsh  combination,  but  it  is  here  duly  prepared,  and 
considering  that  it  occurs  at  the  end  of  a  period  the  effect  is  highly 
satisfactory. 

A  few  quotations  are  included  as  additional  illustrations.     The 
first  two  are  from  Beethoven  : 

Allegretto.    Op.  31,  No.  1. 


Ex.  560. 


r=v? — » — * — *  *  {    *     *  *     f     f 

^ 


The  gth  disappears  in  the  essential  harmony,  as  usual.  In  the  next 
example  the  gth  chord  is  more  fully  represented  and  of  longer  dura- 
tion. Its  disappearance  into  the  dominant  yth  harmony  is  effected 
in  the  last  measure  by  the  simple  omission  of  the  gth : 


GOODRICH'S  ANALYTICAL  HARMONY. 


i  5    3 


The  dissonant  combination  here  appears  uninverted,  though  the  3d^ 
5th,  7th  and  gth  exchange  places  above. 

In  the  excerpt  from  Nicode  the  major  gth  appears  as  a  funda- 
mental harmony  at  (b)  and  at  (c)  : 

Nicodf.    Op.  26. 


Ex.  562. 


- 


* 


X  -Jf- 


An  unusual  example  is  here  quoted  from  an  American  song.  A 
principal  gth  chord  is  resolved  indirectly,  constituting  a  species  of 
avoided  cadence: 

FCERSTEK.     Op.  30,  NO.  1. 


Ex.  563. 


Without  the  gth  this  would  be  equivalent  to  the  3d  resolution  of  a 
dominant  7th  on  ^^a/.  A' fuller  effect  is  obtained  by  including  the 
major  gth,  and  this  remains  as  a  connecting  note  to  the  F-minor  chord. 
The  last  example  illustrates  the  employment  of  an  altered  gth  chord 
of  the  secondary  species : 


GOODRICH 'S   ANALYTICAL    HARMONY. 


BEETHOVEN:    Adagio.    Op.  31,  No.  2. 


Ex.  564. 


It  should  be  remarked  that  this  gth  chord  is  not  an  independent  har- 
mony, but  results  from  the  suspension  of  b  and  f.  These  two  notes 
resolve  as  they  would  in  the  diminished  chord.  By  performing  this 
in  several  other  scales  it  will  be  sufficiently  understood. 


GOODKICH  S   ANALYTICAL   HARMON^, 


PART  XII. 


Chapter  LI. 


SUSPENSION.    THE  THEORY  ILLUSTRATED. 


"  This  arises  through  th«  delaying  of  a  progression  of  a  voice,  which  is  expected  at  a  defi- 
nite time,  or  even  necessary,  and  in  such  a  manner  that  the  voice,  which  has  to  progress  one 
degree  downwards,  in  order  to  occupy  its  position  in  the  following  chord,  lingers  still  upon 
the  tone  of  the  first  chord,  while  the  others  progress  to  the  second,  and  this  voice  does  not 
pass  over  into  the  harmony  until  later." — E.  F.  Richter. 

THIS  is  quoted  to  show  the  ordinary  explanation  of  an  important 
and  interesting  subject. 

Suspension  refers  more  particularly  to  some  part  of  a  chord  that 
is  held  back  while  the  other  parts  move  to  another  harmony.  The 
suspended  tone  thus  forms  a  dissonance,  as  it  does  not  ordinarily 
belong  to  the  second  harmony. 

The  resolution  of  a  suspended  tone  is  the  same  as  it  would  have 
been  had  no  suspension  taken  place. 

SUSPENSION  IN  TWO-PART  HARMONY. 

The  following  two-part  cadence  will  illustrate  this  in  the  simples* 
manner : 


Ex.  565. 


These  parts  suggest  the  tonic  and  dominant  harmonies.  Suppose 
when  the  contralto  moves  to  the  dominant  chord  at  (b),  that  the 
soprano  note,  g.  is  held  back  until  after  the  change  in  harmony 
takes  place : 


GOODRICH'S  ANALYTICAL  HARMONY. 


243 


Ex.  566. 


The  principles  of  resolution  teach  that  a  yth  naturally  resolves  to  a 
6th,  and  that  the  inversion  of  this  is  treated  in  the  same  manner;  i.  <?., 
the  2d  resolves  to  a  3d  : 


Ex.  567. 


7th,   6th,    2d,    3d. 


The  contralto  part  remains  upon  a  until  the  soprano  part  sounds /- 
sharp,  and  all  the  requirements  of  preparation  (a),  suspension  (b), 
and  resolution  (c)  are  fulfilled : 


Ex.  568 


•B 

Ti     - 

^  

(2 

rt 

r^ 

The  g  here  descends  to  f -sharp,  as  it  did  in  the  original  Ex.  565 ;  the 
principal  difference  is  that  this  progression,  g  to  f -sharp,  is  delayed 
until  after  the  main  accent,  where  the  change  in  harmony  is  naturally 
expected  to  take  place.  This  difference,  though  of  primary  impor- 
tance here,  is  not  the  only  one  to  be  noted ;  for  in  a  larger  sense  the 
suspension  produces  a  discord,  and  this  results  in  three  harmonies  in 
place  of  two : 

J. 

Ex.  569. 


.&- 


The  student  should  invert  this  exercise  in  order  to  understand  its 
different  phases.  Simply  write  the  soprano  part  an  octave  lower  for 
contralto  and  let  the  original  contralto  part  remain  as  soprano  part. 

This  is  preferable  to  the  suspension  of  the  3d,  which  results  in 
the  ambiguous  interval  of  a  4th  : 


Ex.  570. 


If  there  were  other  parts  above  or  below  the  4th,  this  plan-  might  be 
adopted,  as  in  this  instance : 


GOODRICH'S    ANALYTICAL    HARMONY. 
Ex.57i- 


244 


But  for  present  purposes  the  dissonant  interval  2,  resolved  to  the 
consonant  3,  is  much  more  desirable. 

As  these  are  the  fundamental  principles,  a  more  extended  applica- 
tion is  in  order,  and  this  model  is  presented  for  elaboration  : 


Ex.  572. 


- 


r 


1 


The  lower  part  is  to  be  suspended  from  the  last  half  of  each  measure.. 

The  student  should  now  work  this  out  to  a  satisfactory  close.  It 
will  end  with  an  authentic,  not  with  an  after  cadence.  *  *  * 

This  design  may  be  freely  inverted  without  the  least  danger  of 
false  progressions,  and  this  is  always  an  advantage.  Write  the  con- 
tralto part  next,  an  octave  higher,  leaving  the  original  soprano  part 
as  it  is.  This  will  change  the  order  of  the  sequence  from  2  —  3  ta 
7 5  *  *  * 

The  first  arrangement  is  here  given  for  comparison  : 


Ex.  573. 


srr^ 


The  exercise  previously  written  by  the  student  should  correspond 
exactly  to  this,  from  which  it  is  a  simple  task  to  arrange  the  inver- 
sion as  suggested. 

SUSPENSION  OF  THE  UPPER  PART  IN  FULL  HARMONY. 

Begin  by  arranging  Ex.  569  in  three  parts,  adding  a  simple  funda- 
mental base.  Then  add  a  middle  part  between  the  soprano  and  con- 
tralto. The  student  is  to  finish  these.  *  *  * 

A  simple  harmonic  design  in  four  parts  is  given,  to  which  sus- 
pensions are  to  be  arranged  in  the  upper  part : 


Ex.  574. 


GOODRICH 'S    ANALYTICAL    HARMONY.  24.3 

The  harmony  is  to  remain  the  same  fundamentally.  Simply  hold 
back  the  upper  e,  and  then  the  d.  Their  final  progression,  e  to  d, 
and  d  to  c,  will  be  the  same  as  in  the  original ;  but  as  a  dissonance 
results  from  each  suspension,  the  e  to  d  will  become  a  resolution, 
rather  than  a  progression.  *  *  * 
The  next  exercise  is  more  extended : 


Ex.  575. 


>;: 


Write  the  suspensions  in  the  upper  part,  beginning  and  ending  in 
G^  Then  add  a  middle  part  between  the  two  parts  already  given. 
\Yhen  the  three  upper  parts  are  completed  the  base  should  be 
added.  Owing  to  the  descending  movement  of  the  treble  parts  this 
will  require  some  care,  if  not  ingenuity.  A  fundamental  progression 
in  the  base  will  be  necessary,  because  the  form  of  the  other  parts  does 
not  invite,  or  even  admit  the  employment  of  real-bases.  After  the 
essential  jth  has  been  introduced  it  will  be  necessary  to  avoid  the 
cadence.  The  suspension  does  not'palliate  such  progressions  as  these, 
which  are  never  allowable  : 


Ex.  576. 


But  the  f-sharp,  a,  d,  above  may  be  treated  as  parts  of  a  secondary 
7th  chord,  and  thus  produce  an  indirect  resolution  corresponding  in 
general  design  to  the  previous  avoided  cadence.  Or  the  B-minor 
chord  may  be  used. 

No  modulation  or  decided  cadence  is  required  in  this  intermediate 
passage.  The  complete  cadence  naturally  follows. 

The  last  of  this  exercise  presents  a  situation  so  different  from 
previous  examples  that  a  few  words  of  explanation  seem  necessary. 
If  the  last  discord  includes  the  3d  in  the  middle  part,  and  the  rule 
of  resolution  be  followed,  the  result  will  be  this  inharmonious  syn- 
chysis : 


*Tbe  complete  harmonization  may  be  made  first,  and  the  suspensions  added  afterwards. 


24*3 


GOODRICH'S  ANALYTICAL  HARMON\. 


Ex.  577. 


The  second  resolved  to  the  unison  is  very  rarely  permissible, — never 
in  such  an  instance  as  this.  Better  keep  the  /#,  as  well  as  the  a,  in 
abeyance,  though  this  is  not  always  effective  : 


Ex.  578. 


The  simplest,  and,  all  things  considered,  the  best  method  is,  to  move 
the  leading-note  down  to  the  dominant,  thus : 


Ex.  579. 


This  leaves  the  last  concord  in  a  complete  form,  and  does  not  destroy 
the  effect  of  the  suspension  above  by  anticipating  its  resolution.  The 
movement  of  the  leading-note  down  a  3d  is  a  progression,  not  a  reso- 
lution ;  but  as  the  a  above  is  kept  in  abeyance  after  this,  (a)  to  (b)  is 
regarded  as  a  partial  resolution,  completed  at  (c).  Though  this  is  a 
perfectly  justifiable  expediency,  the  fact  remains  that  the  descending 
movement  of  a  leading-note  in  a  final  cadence  lacks  decision  and 
completeness.  Compare  (a)  with  (b)  : 


Ex.  580. 


Therefore  some  good  reason  should  appear  as  a  justification  of  the 
progression  at  (a),  at  least  in  terminal  passages. 
The  complete  example  is  given  : 


GOODRICH'S  ANALYTICAL  HARMONY. 

fc=d=p 


247 


The  measures  lettered  (d)  and  (e)  admit  of  different  arrangement ; 
but  if  a  secondary  yth  chord  be  substituted  for  the  B-minor  triad,  the 
£,y#,  a,  coming  after  the  E-minor  chord,  will  strongly  suggest  d$ 
in  the  melody.  At  (e)  C-major  may  be  substituted  for  the  inverted 
A-minor  chord.  Both  methods  should  be  used  as  a  practice,  and  the 
different  effects  particularly  noted. 

It  is  to  be  observed  of  all  these  suspensions  that  the  upper  part 
becomes  a  dissonance  soon  as  the -harmony  changes  to  a  chord  of 
which  the  delayed  tone  forms  no  part,  or,  to  which  it  does  not  natu- 
rally belong.  This  concludes  the  synthetical  exercises. 

The  next  example  consists  of  a  few  simple  chord  progressions. 
These  are  to  be  arranged  with  suspensions  in  the. upper  part : 


Ex.  582. 


All  the  suspended  notes  here  are  to  resolve  down  a  whole  or  a  half 
step  according  to  the  model,  and  to  the  signature.  In  every  instance 
the  delayed  note  is  to  be  a  dissonance.  * 

The  note  that  follows  a  suspension,  when  it  is  3d  or  5th  of  a  chord, 
is  to  be  omitted  from  the  other  parts.  An  instance  of  this  was  given 
in  the  2d  resolving  to  the  unison.  Even  such  arrangements  as  these 
are  not  good,  and  should  be  avoided : 


Ex.  583. 


248 


GOODRICH'S  ANALYTICAL  HARMONY. 


The  effect  is  unnecessarily  discordant  and  confusing.     The  next  is 
much  better : 


Ex.  584. 


These  suspensions  have  their  due  effect,  because  the  resolutions 
are  not  interfered  with.  This  does  not  apply  to  root-tones,  which 
are  nearly  always  doubled.  And  even  such  instances  as  occur  in 
Ex.  581  are  allowable,  because  the  base  is  removed  v'from  the  reso- 
lution a  distance  of  two  octaves.  The  3d  is  used  as  a  real-base  to 
prevent  a  false  progression,  and  on  account  of  C  being  the  sub- 
dominant. 

Supposing  the  last  exercise  to  have  been  completed,  the  student 
may  compare  his  work  with  this : 


Ex.  585. 


Inverted  bases  are  not  here  employed  after  the  suspensions  begin, 
and  this  plan  should  be  followed  as  often  as  possible  until  the  sub- 
ject is  mastered. 

Transpose  the  theme  of  Ex.  582  into  D-flat  and  E-flatt  and  arrange 
in  the  same  manner. 


A^         ^ 

•^      » 

£}          r>}'  —  ' 

J              'f  —  ' 

1  '  b^ 

—  I  '  —  1 

\M7       ~&P 

x 

y 

—  —  ^  

^^.  (5*  

-(S*  5)  — 

-d  —  ^  —  1 

tr 

•ST 

* 

f\*       i 

>      Jj 

—  »  — 

GOODRICH'S  ANALYTICAL  HARMONY. 


249 


Chapter  LII. 


SUSPENSION  CONTINUED. 

SUSPENSION  IN  THE  MIDDLE  PARTS. 

THK  principles  explained  in  the  last  chapter  may  be  applied  to 
any  part. 

AN   EXAMPLE   OF   SUSPENSION   IN  THE  TWO   MIDDLE   PARTS. 


Ex.  586. 


The  same  directions  are  observed.  Thus,  while  g  is  resolving  to 
f-sharp  the  latter  note  is  omitted  from  the  D  chord.  When  the  note 
above  the  root  of  a  chord  is  held  back,  the  root  may  appear  in  the 
base  as  fundamental. 


SUSPENSION  IN  THE  BASE. 


This  is  not  generally  effective  unless  the  base  is  a  solo  or  obligate 
part.     A  few  illustrations  are  appended  for  examination  : 


Ex.  587. 


3^ 


The  tonic  is  first  suspended  and  then  resolved  down  to  the  3d  of  the 
essential  discord,  that  note  being  omitted  from  the  upper  parts.  The 
3d  of  the  tonic  is  next  delayed  in  its  progression,  and  resolves  to  e 
after  the  remaining  tones  of  the  essential  discord  have  been  sounded 
above.  The  theme  being  in  the  base,  that  part  assumes  more  freedom 
and  independence.  Re-arrange  Ex.  587  in  this  position, 


GOODRICH 'S    ANALYTICAL    HARMONY. 


Ex.  588. 


with  the  same  base,  and  transpose  both  arrangements  into  C  and 
B-jlat. 

All  the  chord  progressions  must  be  as  correct  as  though  the  sus- 
pensions did  not  appear.  To  test  this,  simply  omit  the  suspended 
note : 


Ex.  589. 


a                             b 

C~\'              •  —  - 

i 

1 

»•  —  (5 

&       a     1    I?5 

gy             I 

S 

r      r    1  i 

(a)  is  the  same  as  (b),  except  that  e  in  the  base  is  held  back  until 
after  the  harmony  has  changed.  The  e  then  descends  to  d,  as  in  the 
second  example. 

UPWARD  RESOLUTION. 

In  the  exercises  thus  far  the  suspension  has  resolved  down  a  2d. 
This  is  the  usual  tendency,  with  exception  of  the  leading-tone.  In 
all  direct  cadences  this  tone  is  resolved  to  the  tonic,  and  the  fact  that 
the  leading-tone  is  suspended  does  not  alter  its  natural  progression. 

The  5th  of  a  dominant  chord  may  also  resolve  to  the  tone  above, 
and  so  may  any  part  of  a  concord.  These  points  are  illustrated  in 
the  next  example : 


Ex.  590. 


J-J- 


j" 


The  ascending  resolutions  are  indicated  by  crosses.  The  f-s/iarp 
ascends  because  that  is  its  natural  tendency  ;  but  in  the  other  in- 
stances the  delayed  note  ascends  in  order  to  complete  the  harmony. 
Observe  the  places  where  d  ascends  to  e,  and  where  c  descends  to  b* 
Transpose  the  last  example  to  F. 


Ordinary  notes  of  connection  are  not  here  considered  suspensions. 


GOODRICH'S  ANALYTICAL  HARMUXY. 
DOUBLE  SUSPENSIONS. 


251 


These  should  usually  be  either  a  3d  or  a  6th  (major  or  minor) 
apart,  resolving  in  parallel  movement ;  or  an  augmented  4th  (and  its 
inversion,  an  imperfect  5th)  resolving  in  contrary  movement.  A 
suspended  3d,  and  a  suspended  6th,  each  resolving  down,  are  given : 


Ex.  591. 


The  intervals  mentioned  possess  a  certain  harmonizing  relationship, 
and  naturally  go  in  pairs.  Therefore,  when  one  tone  of  the  interval 
of  a  3d  or  6th  is  suspended,  the  other  may  also  be  delayed  in  its  pro- 
gression so  as  to  accompany  its  reciprocal  tone.  The  double  suspen- 
sion is  sometimes  more,  and  at  other  times  less  dissonant  than  the 
single  suspension.  In  the  latter  the  melody  is  more  isolated  and 
independent  than  where  the  delayed  tone  is  accompanied  with  another 
delayed  tone  in  some  other  part.  When  the  leading-note  is  suspended 
it  is  proper  to  delay  the  resolution  of  the  subdominant  also,  as  the 
simultaneous  disappearance  of  these  two  elements  of  transition  is 
very  natural  and  agreeable  : 


Ex.  592. 


Example  (a)  is  preferable  to  (b),  though  the  latter  is  correct.  Unless 
the  base  is  a  solo  part  it  is  better  to  give  it  a  regular  fundamental 
position  as  in  Ex.  591.  Re-arrange  and  transpose  examples  591  and 
592  until  they  are  thoroughly  understood. 

SUSPENSION  RESOLVING  TO  A  CHANGING  HARMONY. 

The  harmony  may  change  at  the  same  time  the  suspension  re- 
solves. In  such  instances  the  note  to  which  the  suspension  resolves 
must  belong  to  both  chords,  and  the  progression  is,  of  course,  to  be 

*The  tied  notes  may  be  repeated  in  order  to  s^ow  the  full  effect. 


252 


GOODRICH'S  ANALYTICAL  HARMONY. 


correctly  managed.  Suppose  b  is  suspended  over  a.  When  the  reso- 
lution takes  place  the  harmony  may  change,  provided  both  chords 
contain  a : 


Ex.  593. 


The  resolution  (b  to  a)  occurs  in  the  full  measure,  and  the  harmony 
changes  from  a  dominant  to  a  diminished  yth  chord.  The  a  belongs 
to  both  harmonies.  Between  these  two  chords  there  are  three  con- 
necting notes.  In  the  next  example  there  are  two  notes  in  common, 
e  and  g : 

B* 

Ex.  594. 


Such  instances  are  always  proper,  provided  care  be  exercised  in 
maintaining  the  connecting  note  in  the  same  voice-part. 

The  resolution  of  the  suspended  a  is  the  same  as  though  the 
E-minor  triad  had  remained  and  not  been  succeeded  by  the  C  chord. 
A  short  theme  here  follows  as  an  exercise  in  suspension : 


Ex.  595. 


^r 

i                           '-^ 

f          ^ 

*?                      —  j—™^ 

.«           ..^       .    1 

rm    I 

1             j 

P' 

{\\j 

1             1 

t 

r  •       !™      1 

+ 

£fc  1— 

f? 

KS 

^-W- 

~\  

This  begins  and  ends  in  C,  with  a  temporary  modulation  to  the  rela- 
tive minor.  Change  the  harmony  during  the  measures  marked  + . 
A  double  suspension  may  be  included  in  the  cadence. 

Transpose  to  A  and  B-flat.  *  *  *  Another  short  theme  is 
given  for  harmonization.  The  student  is  merely  to  supply  the  middle 
parts.  The  bases  are  all  fundamentals  except  in  the  sixth  measure, 
where  the  5th  is  below  : 


GOODKICll'S    ANALYTICAL    HARMONY. 


253 


Ex.  596. 


The  only  difficulty  anticipated  is  in  the  proper  treatment  of  the  ca- 
dence.    It  should  be  written  in  one  of  the  following  ways  • 


Ex.  597. 


-»-     g  I       <g  J 


The  arrangements  at  (a)  and  (c)  are  best.    After  completing  Ex.  596, 
transpose  into  F  and  A. 

PREPARATION  BY  MEANS  OF  A  SINGLE  TONE. 

Heretofore  the  preparation  of  the  delayed  tone  has  been  accom- 
plished by  means  of  a  chord  in  which  the  suspension  formed  a  con- 
sonant interval.  This  is  not  always  necessary.  It  is  sufficient  if  the 
single  tone  that  produces  the  dissonance  was  heard  previously  as  a 
regular  harmonic  tone.  The  example  illustrates  this : 


Ex.  598. 

J=? 


3ES 


r 


|=^— T 


The  first  melodic  note  in  the  second  measure  is  little  more  than  a 
prepared  appoggiatura,  but  the  same  principle  applies  here.  It  serves 
as  a  preparation  to  the  double  dissonance  in  the  second  and  fourth 
measures.  In  addition  to  these  tied  notes  (b  and  g)  the  impression 
of  tonic  harmony  is  created  by  the  chord  figures  in  the  first  and  third 
measures,  and  this  impression  still  lingers  until  the  essential  discord 
is  fully  developed. 


254 


GOODRICIl'S    ANALYTICAL    HARMONY. 
SUSPENSIONS  ON   INVERTED  CHORDS. 


An  illustration  follows  in  which  the  suspensions  are  accompanied 
with  inverted  bases.  Extreme  care  is  necessary  in  this  matter,  and 
in  addition  to  the  preliminary  example  the  student  is  advised  to  seek 
illustrations  from  standard  composers  : 


»  9  *- 

i  —  b?— 

^ 

^ 

Jd*  L 

^\)                &             ^fS          1 

g_^j? 

^ 

I 

^                  1 

Ex.  599. 

J      r     1 

C\*   |-          .            1 

1 

~zJ'ti^  d  ^  

—  ^  



r                  ^^ 

^ 

^f 

^             ' 

•      , 

|  1                     I 

i     , 

k_j 

• 

JT    «        £* 

jJ^T                                   ^^                                   ^"^ 

Jj^        A 

Bl 

£r<                  £r 

RTr      L/l 

^                 "fl                                                          ^^ 

iL 

-d  —  ^^- 

-i  1 

- 

1-                                    -f 

r^             ^ 

<^M 

^4 

<5^  — 

Q-ti2— 

=3  f— 

3=1=1 

All  these  chord  progressions  are  treated  exactly  as  though  the  sus- 
pensions did  not  exist.     Delayed  notes  occur  alternately  in  each  of 
the  four  voice-parts.    The  real-base  in  every  instance  is  to  be  omitted 
above,  as  it  supplies  a  void  in  the  upper  harmony. 
The  last  example  should  be  written  in  B  and  C. 

SUSPENSIONS  UNRESOLVED. 

There  are  instances  in  which  the  suspended  tone  is  not  resolved 
to  the  tone  above  or  below,  but  either  remains  as  part  of  the  following 
harmony,  or  skips  to  the  tonic  of  the  last  chord.  An  example  of  the 
latter  is  quoted  from  the  score  of  "  Cavalleria  Rusticana": 


Mo.sco.gni. 


Ex.  600. 


The  suspended  j^  descending  to  the  tonic,  e,  is  a  melodic  license. 
The  composer  simply  omits  the  f-sharp  between  g  and  e.  The 
procedure  i*  unusual,  but  the  effect  is  sufficiently  characteristic  to 
justify  it. 


GOODRICH'S  ANALYTICAL  HARMONY. 


255 


The  other  instance  in  which  the  suspended  dissonance  (or  ap- 
poggiatura)  is  retained  and  becomes  part  of  the  ensuing  concord 
has  been  used  by  several  composers.  A  good  illustration  may  be 
found  in  Kamennoi  O straw,  by  Rubinstein.  It  occurs  in  the  middle 
part,  immediately  before  the  cadenza. 

In  all  serious  music  these  different  kinds  of  suspension,  and  the 
resolutions  resulting  therefrom,  are  of  great  importance,  and  furnish 
a  considerable  amount  of  adventitious  aid  to  the  art  of  counterpoint. 
They  also  serve  as  connecting  links,  even  where  the  harmonies  are 
entirely  dissimilar,  and  what  is  more  important,  they  add  great  va- 
riety to  the  harmonic  coloration;  for  the  number  of  combinations 
possible  by  means  of  suspension  is  innumerable. 


Chapter  LIII. 


PEDAL-NOTE.    (ORGAN-POINT.) 


TONIC  PEDAL-NOTE. 


THIS  refers  to  a  stationary  tone  in  the  base,  upon  which  various 
related  and  unrelated  harmonies  are  sounded. 
The  tonic  is  best  adapted  to  support  these  changing  harmonies, 
"because  several  chords  are  generated  from  this,  when  considered  as 
a  fundamental. 

The  simplest  harmonic  material  for  a  pedal-note  passage  com- 
prises those  chords  which  contain  the  tonic.  There  are  three  of 
these  in  every  major  and  minor  scale,  thus: 


Ex.  601. 


256 


GOODRICH  S    ANALYTICAL    HARMONY. 


The  pedal-note  forms  a  constituent  part  of  each  chord,  and  the  basis 
is,  therefore,  a  perfectly  natural  one.  <,The  student  should  write  a 
similar  example  in  minor.)  *  *  * 

The  dominant  chord  is  not  immediately  connected  with  the  note 
of  the  tonic;  but  we  have  seen  that  the  dominant  7th  chord  ma 
suspended  above  the  tonic  by  means  of  retardation.  The  stationary 
base  is  a  sufficient  preparation  of  the  dissonance,  though  there  may 
be  a  still  farther  connection  between  the  dominant  chord  and  its  ante- 
cedent. This  added  to  the  last  example  will  comprise  the  principal 
material  for  nearly  all  pedal-note  passages : 


Ex.  602. 


i   * 


Tonic  Pedal. 


* 


sfe 


All  the  chords  are  connected  with  the  pedal-note,  excepting  at  — . 
This  is  merely  connected  through  the  fifth  of  the  tonic.  The  sepa- 
rate links  may  thus  be  formed  into  a  chain  of  harmonies : 


Ex.  603. 


g=g 


2- 


~  g — « — 


As  these  are  simple  chord- products  of  the  scale  of  C,  and  as  they  are 
all  connected  in  their  progression,  the  fundamental  of  the  scale  max- 
well serve  as  permanent  tonal  foundation. 

We  have  thus  far  the  governing  principles  for  organ-point  in 
general.  By  observing  these  principles  it  will  be  comparatively  easy 
to  elaborate  such  designs  as  the  last. 

We  have  ascertained  that  the  dominant  7th  chord  may  be  sounded 
upon  the  tonic  by  means  of  suspension  ;  and  according  to  this  doc- 
trine almost  any  combination  may  be  accounted  for.  The  main  con- 
ditions are  that  the  chords  progress  smoothly  and  without  any  de- 
cided transition  to  a  strange  tonality.  The  following  is  a  simple 
illustration : 


GOODRICH'S  ANALYTICAL  HARMONY. 

Andante. 


257 


Ex.  604. 


i        I 


The  extract  begins  and  ends  with  the  tonic  chord,  of  which  the  pedal- 
note  is  root. 

The  pedal-note  can  occur  any  where  during  the  progress  of  a 
composition,  but  its  usual  place  is  at  the  close.  Frequently  it  is  the 
foundation  of  a  coda. 

Eight  measures  of  the  Sanctus  from  Haydn's  Imperial  Mass  are 
here  quoted.  This  occurs  after  the  complete  cadence,  and  forms  a 
codetta  to  the  movement  : 


Ex.  605. 


The  harmonic  scheme  is  perfectly  simple,  but  none  the  less  effective : 
It  begins  on  the  tonic,  followed  by  a  passing  modulation  to  the  sub- 
dominant,  and  then  a  transition  back  to  D  by. means  of  the  principal 
diminished  chord  resolved  to  tonic  major.  The  main  advantages 
of  the  pedal-note  are  to  be  found  in  the  unity  and  tenacity  which  it 
imparts  to  the  passage.  The  sentiment  is  Hosanna  in  excelsis. 

DOMINANT  PEDAL-NOTE. 

The  next  example  illustrates  a  dominant  pedal-note  against  chro- 
matic passing  chords  above : 


Nicodi. 


Ex.  606. 


GOODRICH'S  ANALYTICAL  HARMONY. 


Melodic  designs  that  require  regular  fundamental  harmonies  as  an 
accompaniment  (such  as  Ex.  523),  and  all  transitions  that  hyve  the 
effect  of  establishing  a  new  tonality,  are  to  be  avoided  in  pedal-note 
passages ;  for  this  stationary  base  precludes  the  possibility  of  a  series 
of  changing  fundamentals. 

To  simple  designs,  such  as  those  of  the  first  four  illustrations,  the 
pedal-note  imparts  something  of  simplicity  and  rustic  charm  ;  to  such 
passages  as  those  quoted  from  Haydn  and  Nicode  the  organ-point 
adds  dignity,  firmness  and  tenacity  of  purpose. 

The  pedal-note  explains  many  combinations  that  would  otherwise 
be  inexplicable.  This  presupposes  that  the  pedal-note,  considered  as 
permanent  foundation,  is  a  harmonic  necessity.  The  following  ex- 
ample from  a  comparatively  recent  text-book  on  Harmony  will  serve 
to  exemplify  this : 

^ 

Ex.  607. 


This  is  unsatisfactory  and  faulty.  After  retaining  G  in  the  base 
during  the  2d  measure  (where  it  becomes  5th  of  the  tonic  chord) 
the  lower  f  produces  an  incongruous  effect.  There  are  two  reasons 
for  this  :  i.  There  is  no  connecting  note  between  the  harmonies  at 
(a)  and  (b) ;  2.  the  real-base,  g,  does  not  naturally  progress  down  to 
f  in  the  present  instance.  But  by  considering  G  as  a  pedal-note  and 
retaining  it  during  the  next  measure,  a  more  musical  and  consistent 
effect  is  produced : 

-I- 


Ex.  608. 


It  is  not  well  to  alter  the  base  in  such  instances  (as  was  done  in  the 
3d  measure  of  Ex.  607),  because  the  effect  is  too  disconnected  and 
incoherent.  By  retaining  the  G  as  pedal-note  a  better  result  is 
obtained,  and  nothing  stranger  than  a  major  9th  appears : 


GOODRICH'S  ANALYTICAL  HARMONY. 


Ex.  609. 


This  presupposes  that  G  was  heard  as  a  real-base  in  the  previous 
measure.  In  the  same  manner  the  subdominant  harmony  may  ap- 
pear upon  the  dominant  pedal  where  the  chord  of  the  dominant 
follows,  thus  : 


Ex.  610. 


Gilchrist. 


It  is  useless,  and  even  absurd,  in  such  instances  to  change  the  base 
to  C  at  (a)  and  then  back  to  D  at  (b). 

The  harmony  of  the  diminished  7th  may  be  freely  used  above  the 
pedal-note.  The  chromatic  character  of  this  chord  is  well  adapted 
to  such  purposes : 


Ex.  611. 


The  tonality  of  G  is  not  seriously  interfered  with  here ;  and  even 
if  the  chromatic  chords  should  have  a  tendency  to  alter  the  key-im- 
pression to  a  certain  listener,  the  pedal  would  serve  to  keep  the  atten- 
tion fixed  upon  this  point. 

The  stretto  to  a  fugue,  usually  founded  upon  an  organ-point, 
affords  the  best  illustration  of  the  general  character  and  effect  of  a 
sustained  fundamental.  The  polyphonic  style  of  a  fugue  excludes 
those  features  previously  mentioned  as  objectionable  here. 

A  good  illustration  is  quoted  from  the  stretto  to  a  fugue  in  G: 


GOODRICH'S  ANALYTICAL  HARMONY.. 


Cherubini. 


The  soprano  and  tenor  answering  each  other  at  the  interval  of  a  5th 
is  in  the  canonic  style;  the  chord  feature  is,  therefore,  not  prominent. 
*  Whether  the  pedal-note  be  located  upon  the  tonic,  or  the  domi- 
nant, its  treatment  remains  the  same.  See  Exs.  606,  611  and  612. 

The  first  and  last  chords  should  be  connected  with  the  pedal-note, 
but  the  intermediate  passages  may  be  related  only  in  a  general  way, 
as  we  have  seen.  When  the  pedal  passage  contains  modulations  to 
the  related  keys,  and  the  harmonic  connection  is  slight,  it  is  still  more 
necessary  that  the  first  and  last  chords  should  be  in  harmony  with 
the  sustained  base. 

As  the  upper  parts  are,  in  their  progression,  independent  of  the 
pedal-note,  the  former  should  be  complete  in  themselves, — though 
the  pedal-note  may  be  relied  upon  to  supply  a  suitable  foundation 
in  lieu  of  the  ordinary  movable  base. 

The  tonic  of  any  scale  to  which  a  decided  transition  is  effected 
may  be  continued  in  the  base,  and  thus  become  a  pedal-note.  In 
such  instances  the  harmony  above  is  to  be  treated  as  though  the 
organ-point  represented  the  key-note,  or  its  dominant.  There  are 
some  exceptions  to  this,  but  none  to  the  statement  that  all  pedal-note 
passages  are  governed  by  the  same  fundamental  principles. 

TONIC  AND   DOMINANT   PEDAL-NOTES  COMBINED. 

The  two  pedal-notes  previously  explained  are  frequently  com- 
bined after  the  manner  of  a  bag-pipe  or  drone-base.  When  the  har- 
mony consists  chiefly  of  tonic  and  dominant,  this  plan  is  very  effect- 
ive. Following  is  a  simple  illustration : 


GOODRICH  S    ANALYTICAL    HARMONY. 


26r 


Ex.  613. 


-^r. N 


r 1 1 ai-H — i 

t=»=it=g±* 


=±t± 


The  double  pedal  assumes  the  character  of  a  fundamental  accompa- 
niment, one  or  other  of  the  pedal-notes  being  connected  with  every 
chord  above.  Beethoven  introduced  several  effects  of  this  kind  in 
his  Pastoral  Symphony.  See  the  allegro,  the  scherzo,  and  the  finale. 
The  double  pedal  may  also  support  chromatic  progressions  as 
passing  harmonies : 


Ex.  614. 
fcrf 


M.  Oestcn. 


^^^MS 


i 


|      ^ 

*—  s  --  * 


The  tonic  chord  occurs  on  the  first  of  each  measure  where  the  strong- 
est rhythmical  accent  naturally  falls.  The  dissonances  are,  therefore, 
less  harsh,  because  less  noticeable. 

The  next  example,  of  a  livelier  character,  is  quoted  from  Bach- 
uiann's  Danse  Bretonne  : 


Ex.  615. 


262 


GOODRICH'S  ANALYTICAL  HAK.MOXY. 


The  harmonic  impressions  created  by  the  melody  are  so  slight,  and 
the  drone-base  is  so  appropriate  to  dances  of  this  kind,  that  the  double 
pedal  serves  as  a  very  natural  foundation. 

A  more  artistic  illustration  of  pedal-note  is  here  extracted  from 
the  Polish  Dances  by  Ph.  Scharvvenka,  Op.  38,  No.  2  : 


Ex.  616. 


:  J"*1  ^^^^"bT^^^.  -i  4^Pi 

IZ-*          r  P*3  •*•  jz-r-*- 


Py7  i       i 

i 

-d  

—  ^h 

•*•  .  -*•  .  •«*•  .  •*•  .  •*•  . 

All  the  intermediate  harmonies  are  unrelated  to  the  pedal-notes, 
though  the  key-impression  of  F  remains  throughout.  This  organ- 
point  adds  somewhat  of  seriousness  to  the  darkly-colored  foreground, 
and  gives  consistency  to  the  constantly  changing  harmonies  above. 

The  second  part  to  rococo  gavottes,  called  musette,  will  be  found 
to  contain  many  examples  of  pedal-note  that  may  be  advantageously 
studied  in  connection  with  this  subject,  for  the  dance  of  that  name  is 
usually  founded  upon  a  single  or  double  pedal  throughout. 

A  volume  of  organ  mtisic  will  furnish  instructive  examples  of 
pedal-note.  (See  also  the  first  thirty-two  measures  of  "  Ophelia,"  by 
E.  Nevin,  Op.  13,  No.  2.) 

Advanced  students  should  consult  orchestral  scores,  for  some  of 
the  most  interesting  pages  of  modern  music  are  built  upon  a  station- 
ary tonal  foundation. 

In  piano  music  the  organ-point  is  not  so  frequently  employed, 
nor  is  it  so  effective,  though  Chopin's  exquisite  cradle  song  is  founded 
upon  a  double  pedal  from  beginning  to  ending.  Even  in  the  coda 
the  tonic  pedal  is  maintained. 


GOODRICH'S  ANALYTICAL  HARMONY. 


263 


Chapter  LIV. 


SEVEN  ADDITIONAL  RESOLUTIONS  OF  THE 
DOMINANT  SEVENTH  CHORD. 


'"pHIS  is  a  continuation  of  Chapters  XXIV  and  XXV,  and  refers 
-*-  to  the  disappearance  of  the  essential  discord  into  a  concord. 
All  these  additional  resolutions  are  indirect,  and  when  they  occur 
at  the  end  of  a  strain  they  constitute  a  species  of  avoided  cadence. 
Otherwise  they  are  mere  progressions.  They  are  numbered  con- 
tinuously from  the  first  four  previously  explained. 

No.  5  is  to  a  major  concord  whose  root  is  the  same  as  the  yth 


of  the  discord  :      Ex.  617.  hJjL    g=an         It  may  be  used  in  almost 

-"       —  ~ 


any  position.     Several  arrangements  are  given  : 


Ex.  618. 


te    Ug=g=zJ 

=5±^=fcJ 


f— f — i pf 


The  yth  remains  stationary  ;  the  root  and  3d  ascend  a  2d  each ;  the 
5th  descends  a  2d.  When  the  root  is  in  the  base  it  commonly  as- 
cends or  descends  to  the  5th  of  the  concord.  See  (a)  and  (f ).  The 
root  in  the  base  may  also  ascend  to  the  3d  of  the  concord,  as  it  did 
in  the  first  example.  See  (d)  and  (e).  The  3d  is  then  omitted  from 
the  upper  parts.  When  the  base  is  inverted  it  resolves  according  to 
the  previous  directions.  See  (b)  and  (c). 

The  resolution  of  b  down  to  a  in  the  last  measure  is  less  natural ; 
but  it  is  necessary  to  prevent  the  last  concord  appearing  blank—- 
without its  3d. 

The  most  common  application  of  this  5th  resolution  is  to  found 
it  upon  the  2d  note  of  the  scale,  and  then,  by  placing  the  5th  of  the 
following  concord  in  the  base,  to  return  to  the  key-tone,  thus : 


264 


GOODRICH'S  ANALYTICAL  HARMONY. 


Rertini. 


Q    .          ;  —                  """"--     .      -"                          "~~~- 

fe?-2__  ! 

-f- 

-f^  —  =  

fc  •.. 

i                                                             i 

•^        •*        •*         V          i          \m        >*        "*•        •*"          V     i   ^~        ^~         V        •*" 

*     -«•-»•      ^     -*••  -*•-    «^     -9-     -9-      f-  Y*r     ir  •  •  *    *W 

P*  u 

*              ^ 

«        » 

?        'f 

* 

•._? 

rTu 

<2?     • 

1          1 

^ 

Ex.  619. 


The  concord  with  its  5th  in  the  base  may  be  obtained  by  giving  the 
root  to  the  base,  and  causing  that  part  to  ascend  a  4th  or  descend  a 
5th,  as  at  (a)  or  (f )  in  the  previous  example.  The  essential  discord 
to  the  tonic  naturally  follows,  as  in  the  last  illustration. 

The  6th  resolution  is  similar  to  this, but  in  the  minor  mode  thus: 


Ex.  620. 


The  5th  of  the  discord  is  omitted  in  the  second  measure,  but  this  is 
of  no  particular  consequence.  The  same  directions  apply  to  this,  as 
5  and  6  correspond. 

Beethoven  used  this  much  more  freely  in  the  development  of  O>> 
7,  just  before  the  reprise : 


Ex.  621. 


The  effect  here  is  that  of  a  harmonic  digression.  The  chord  marked 
peonies  so  unexpectedly  that  it  forces  the  attention  away  from  the 
tonality  of  A-minor  in  a  most  emphatic  manner.  This  leads  to  D-mi* 
nor,  as  might  be  presumed  from  the  base. 

No.  7  resolves  to  a  minor  triad  whose  root  is  the  5th  of  the  dis- 
cord. This  note  must  not,  therefore,  be  omitted.  There  are  two 
connecting  tones,  5th  and  7th  of  the  discord,  while  the  root  and  3d 
both  resolve  to  the  5th  of  the  concord. 

The  following  positions  are  available  : 


GOODRICH'S  ANALYTICAL  HARMONY. 

i**?-— 


265 


Ex.  622. 


These  are  all  resolved  in  the  same  manner,  because  no  other  alter- 
native presents  itself.  But  in  the  progression  following  the  yth  reso- 
lution considerable  liberty  may  be  allowed!  Advantage  can  be  taken 
of  the  I  chord  in  order  to  pass  to  D-minor  by  introducing  the  domi- 
nant ;  or  consider  the  D-minor  triad  as  a  mere  passing  chord  in  pro- 
gression. Examples  of  these  two  methods : 


Ex.  623. 


Others  are  possible,  but  the  author  gives  the  most  feasible,  and  leaves 
the  ingenious  young  composer  to  the  pleasures  of  discovery. 

The  8th  resolution  is  likewise  to  a  minor  triad.    The  root  and  3d 
remain ;  the  5th  and  yth  resolve  up  and  down  to  the  root  of  the  triad : 


Ex.  624. 


m 


This  is  a  mere  progression.  Its  most  peculiar  feature  is  that  it  admits 
of  inversion  more  readily  than  do  the  others.  In  truth,  the  uninverted 
form  is  not  very  serviceable. 

No.  9  is  more  transitional.  So  far  as  the  author  has  been  able  to 
discover,  it  was  first  employed  by  Schubert  in  his  great  C-major  Sym- 
phony. The  root  of  the  discord  remains  as  3d  of  the  major  concord  ; 
the  3d  descends  a  chromatic  step  ;  the  5th  ascends  a  minor  2d,  and 
the  yth  descends  a  major  2d,  to  the  root  of  the  concord : 


Ex.  625. 


266 


GOODRICH'S  ANALYTICAL  HARMONY. 


It  is  the  most  abrupt  of  all,  excepting,  perhaps,  No.  10.  In  the  ex- 
ample from  Schubert  the  5th  of  the  discord  is  real-base,  and  resolves 
directly  to  the  root  of  the  triad  : 


Ex.  626. 


TTI 


This  is  much  more  effective  than  the  uninverted  form.  Observe  that 
the  root,  3d  and  yth  of  the  discord  are  duplicated  above  by  the  differ- 
ent instrumental  parts,  but  that  the  5th  is  given  only  to  the  bases. 
The  immediate  effect  is  bright,  and  rather  bold,  the  concord  on  E-JJat 
being  altogether  unexpected.  This  extract  is  taken  from  the  2d  sub- 
ject of  the  first  allegro. 

No.  10  is  also  novel,  but  perfectly  feasible : 


Ex.  627. 


10. 


It  is  better  to  omit  the  3d,  as  it  has  no  natural  resolution  here. 
following  positions  are  perhaps  best  adapted  to  practical  uses  : 


The 


Ex.  628. 


The  immediate  resolution  is  remote,  and  might,  for  this  reason,  be 
utilized  in  making  a  distant  transition,  as  in  the  last  example  (c). 
This,  however,  is  but  one  of  the  many  ways  in  which  the  harmony 
might  be  directed  after  the  loth  resolution  has  taken  place. 

The  nth  and  last  of  the  consonant  resolutions  of  a  dominant  yth 
chord  is  to  the  major  triad  located  a  whole  step  above  the  root  of  the 
former : 


;OODRICH'S  ANALYTICAL  HARMONY. 


207 


This  is  not  a  very  natunl  termination  of  the  discord,  but  in  certain 
progressions  it  might  be  utilizedj  as  here  : 


z=z — * — * — •- 


Ex.  630. 


The  end  justifies  the  means.  It  is  well  to  know,  however,  that  this 
resolution  is  abrupt,  even  in  the  midst  of  the  chromatic  progression, 
and  should  not  be  used  ordinarily.  Nearly  all  these  examples  can 
be  reversed,  as  were  numbers  623  and  624.  (Another  resolution  is 
possible.) 

Some  of  the  preceding  resolutions  the  author  has  given  on  his  own 
responsibility.  Less  than  one  half  have  appeared  in  the  text-books, 
but  most  of  the  others  have  been  employed  by  living  composers. 

In  adding  to  those  already  known  the  main  object  has  been  to 
introduce  a  greater  variety  of  harmonic  progressions,  for  we  are 
naturally  disposed  to  employ  only  those  sanctioned  by  frequent 
usage.  But  the  cadence  forms  have  been  used  as  the  harmonic 
basis  of  so  many  thousands  of  songs  and  instrumental  pieces  that 
they  no  longer  interest  a  cultured  listener,  and  almost  any  reason- 
able progression  is  a  relief  from  their  satiating  effect. 

The  substance  of  this  chapter  is  to  be  worked  out  by  the  student 
in  at  least  four  other  scales.  It  would  also  be  well  to  make  a  diagram 
of  the  entire  eleven  resolutions  in  some  key  different  from  that  of  the 
printed  examples. 


268 


GOODRICH  S   ANALYTICAL    HARMONY. 


PART  XIII. 


Chapter  LV. 


DUPLICATION  AND  OMISSION. 

DUPLICATION. 

A  LIMITED  knowledge  of  these  subjects  has  been  acquired  from 
A  Chapters  XXII,  XXVII,  XXIX  and  L,.  It  is  well  known  that 
the  root  of  a  chord  admits  the  greatest  amount  of  duplication,  and  as 
that  tone  is  the  foundation  of  the  chord  this  requires  no  farther  ex- 
planation than  the  following : 

Brahms.  Dvov&k. 


Ex.  631. 


^B 

-ig     '  ••& — J 
->*»-       •)*- 


This  principle  applies  to  all  fundamental  harmonies. 

The  major  3d  is  of  a  decided  character,  and  its  appearance  in  the 
last  example  shows  that  it  will  not  admit  the  same  amount  of  dupli- 
cation as  does  the  root. 

In  harmonic  masses  the  3d  may  appear  in  each  group : 


GOODRICH  S    ANALYTICAL    HARMONY. 


269 


In  an  orchestral  score  this  would  be  perfectly  satisfactory,  for  there 
are  five  r's  against  three  <?'s.  But  when  the  major  3d  appears  as  real- 
base  it  is  usually  omitted  from  the  other  parts.  There  are  two  rea- 
sons for  this :  i .  The  blank  above  is  most  agreeable  filled  by  the 
real-base : 


Ex.  633. 


±1: 


2.  If  the  real-base  is  doubled  above,  it  is  liable  to  produce  parallel 
octaves.  But  where  the  duplicated  interval  progresses  in  contrary 
or  oblique  movement  it  may  be  freely  admitted : 


Ex.  634. 


•# — Mr* — » — 0- 


At  +  in  the  first  measure  the  major  3d  is  doubled,  but  in  the  next 
chord  one  e  descends  while  the  other  remains.  In  the  second  meas- 
ure the  duplicated  minor  3d  disappears  contrarily.  Both  examples 
are  good. 

Another  exception  occurs  when  the  base  passes  through  the  dif- 
ferent notes  of  a  chord,  as  here : 


Ex.  635. 


Or,  where  the  base  skips  to  and  from  the  root,  as  in  this  from  Men- 
delssohn : 


Ex.  636. 


270 


GOODRICH'S    ANALYTICAL    HARMONY. 


This  gives  to  the  base  a  more  individual  character,  and  the  dupli- 
cated 3d  above  remains  passive.  Notwithstanding  these  exceptions 
(which  are  proper  only  when  properly  applied)  the  fact  remains  that 
the  first  measure  in  the  next  example  is  more  satisfactory  than  the 
second : 


Ex.  637. 


n                | 

4 

J 

\J  *f^     5^ 

|     , 

99 

»«>    ez: 

^a 

**—  '  U    ^i 

<s 

S 

W^^       Tl            ^X 

-5* 

1    ^  ' 

^S 

«*       «< 

^      1 

v-  iy 

* 

0 

«y            ^ 

1 

+ 

-t- 

p            -0-                & 

1                + 

It  is  better,  therefore,  not  to  double  the  3d  except  under  such  circum- 
stances as  have  been  enumerated.  The  same  directions  will  apply 
to  the  minor  3d,  though  it  is  not  so  strong  as  the  major  3d.  If  the 
melody  does  not  prevent,  it  is  usually  better  not  to  double  the  real- 
bases  in  such  passages  as  these  : 


Ex.  638. 


This  plan  has  the  advantage  of  greater  purity  to  recommend  it. 

The  5th  admits  of  duplication  more  readily  than  does  the  3d,  espe- 
cially in  cadences  where  the  real-base  becomes  a  fundamental : 


Ex.  639. 


Although  there  are  four  a's  against  one  d  and  one  /"#,  the  effect  is 
satisfactory.  The  real-base  assumes  something  of  the  character  of  a 
fundamental,  and  may  thus  support  the  chord  above  without  causing 
the  duplicated  tones  to  sound  unpleasantly  prominent.  But  where 
the  second  inversion  is  not  followed  by  the  dominant  harmony  the 
5th  should  not  be  doubled,  except  for  a  good  reason.  The  next 
example  may  serve  as  a  model: 


GOODRICH'S  ANALYTICAL  HARMONY. 


771 


i=^=i= 

±-i-l  *7^= 

r=» — 4 — **^"i 


Ex.  640. 


None  of  the  real-bases  are  here  duplicated.  Where  each  of  the  ex- 
treme parts  have  a  melodic  progression  in  contrary  movement  the 
5th  may  be  doubled  above : 

Hummel. 


Ex.  641. 


A  sufficient  reason  for  the  duplication  is  here  apparent.     All  such 
examples  are  correct.     The  next  is  similar: 


642. 


ttj  T—  1  ' 

1  1— 

1  1  \~^~\ 

rm/  ZIEJ       « 

-Ui*         «'  ""^   1 

x#* 

«       m 

i  jl^      5         i 

J         ^JT   ^ 

•     i     £ 

«2zp 

r-v«    i. 

*-JT7  ?       - 

2E  b  '-L 

i 

/   7      *        ^ 

SJ       J 

S*       * 

K  

—  T  —  ^~ 

-J  —  i— 

!          ^ 

(u)-  ^   % 

• 

•J                              *     ~ 

f   ~jf 

\      l 

•»•    -r 
1       I 

r™x  •      i 

i 

-is  f 

-^  b  -i      i              * 

, 

J   J         1    -  -  m 

* 

_i           ' 

-\           *                             - 
^^                           (S>     n* 

^,    * 

In  the  last  measure  the  5th  is  doubled,  even  to  the  exclusion  of  the 
3d ;  but  the  progression  of  ascending  thirds  against  the  descending 
base  justifies  this  procedure.  Of  the  same  nature  is  the  following 
more  artistic  design  quoted  from  Beethoven  : 


Ex.  643. 


FINALE.    Op.  2,  No.  3. 


272 


GOODRICH  S    ANALYTICAL    II AKMON  Y. 


The  fifths  and  thirds  are  here  duplicated  with  the  greatest  of  free- 
dom, though  this  is  the  result  of  expediency. 

The  inversions  of  a  dominant  yth  chord  should  not,  however,  be 
doubled,  unless  some  particular  melodic  design  justifies  the  dupli- 
cation. 

The  next  example,  in  any  of  its  re-arrangements,  may  be  taken 
as  a  safe  model . 


Ex.  644. 


(1) 


The  jth  may  be  doubled  in  the  base,  but  not  in  the  upper  parts. 
Such  unison  passages  as  these  are  perfectly  proper  : 


Ex.  645. 


Z=P= 


1 


(3)  (1) 


Particular  attention  is  directed  to  the  third  inversion  of  the  essential 
discords  in  the  first  and  third  measures. 

These  directions  do  not  apply  to  diminished  yth  chords  except 
where  they  are  treated  as  principal  discords.  When  they  appear  in 
a  secondary  capacity  as  passing  chords,  or  in  chromatic  progressions, 
they  are  not  subject  to  the  same  restrictions,  for  any  note  of  a  dimi- 
nished yth  chord  may  be  considered  root,  3d,  5th  or  7th,  according 
to  its  notation.  Correct  progression  of  the  parts  should  determine 
whether  a  certain  note  may  or  may  not  be  duplicated ;  for  neither 
the  root,  3d,  5th  nor  7th  possess  that  distinctive  tonal  character  which 
is  recognized  in  the  dominant  7th  chord. 

The  same  principle  applies  in  a  modified  sense  to  the  secondary 
7th  chords.  Such  instances  as  the  following  are  of  frequent  occur- 
rence, though  care  must  be  exercised  to  avoid  false  progressions : 


GOODRICH'S  ANALYTICAL  HARMONY.  273 

JL 


Ex.  646. 


Transpose  and  re-arrange  all  the  examples.     *    *    * 

OMISSION. 
The  root  of  a  concord  can  not  be  omitted  without  destroying  its 

identity      EX.  647.  F  j^  —  3      The  latter  is  simply  a  binary 

mj  _~^i~  ^ 

+ 
chord  on  <?.     With  discords  the  same  principle  prevails 

Ex.  648 


In  each  instance  the  nature  of  the  discord  changes  with  the  disap- 
pearance of  its  root,  and  different  treatment  is  required.  Some  theo- 
rists claim  that  the  discord  on  G-sharp  is  a  principal  gth  "  with  the 
root  omitted."  This  is  as  unreasonable  as  to  assert  that  three  right 
angles  form  a  square.  The  diminished  jth  chord  is  used  in  compo- 
sition without  regard  to  a  supposed  root  below  the  theoretical  foun- 
dation of  the  harmony,  and  it  seems  to  the  author  like  chasing  phan- 
toms to  call  on  the  imagination  in  a  matter  that  is  based  entirely 
upon  practice. 

To  return  to  the  concords.  The  3d  is  seldom  omitted  from  a 
major  or  a  minor  triad,  because  the  root  and  5th  sound  blank  and 
equivocal.  As  a  general  rule  it  can  not  be  determined  whether  the 
chord  is  major  or  minor,  and  in  such  instances  the  effect  is  usual!'; 
unsatisfactory  • 


Ex.  649. 


•2-1 J  GOODRICH'S  ^ANALYTICAL    HARMONY. 

Such  an  arrangement  is  scarcely  conceivable  in  a  standard  composi 
tion.  In  two  or  three-part  writing  the  3d  of  the  dominant  chord  maj 
be  omitted  when  it  is  preceded  by  the  tonic  harmony : 


Ex.  650. 


$T^~*~ 

& 

£/ 

~-£  —  ~zr 

^T\ 

—  %~ 

~£L-\ 

a 

'g* 

b 

19- 

cy  —  1  (2 

1  1         /5— 

\ 

5*1— 

o- 

1  1  ^  

-     —  j         I  



In  the  first  phrase  the  key  of  C-major  is  at  once  recognized,  and  there 
is  no  doubt  as  to  the  major  character  of  the  dominant  chord.  The 
same  is  true  of  the  blank  chord  at  (b)  in  the  A-minor  phrase.  But 
this  would  not  be  advisable  in  four  or  five-part  harmony,  because 


the  progression,      EX.  651. 


is  really  an  expedi- 


ency  that  occurred  originally  in  a  duet,  and  most  probably  between 
two  natural  horns:  For  the  tone/"  in  the  following  example  is  not 
one  of  the  open  sounds,  and  the  composer  was,  therefore,  obliged  to 
choose  the  form  at  (a),  instead  of  the  one  at  (b)  : 


Horn-    In  I   . 


Ex.  652. 


Care  must  be  taken,  however,  that  the  blank  chord  will  be  recognized 
as  the  dominant. 

In  a  final  cadence  the  tonic  chord  frequently  appears  without  its 
rth,  because  in  certain  positions  of  the  essential  discord  there  is  no 
note  that  will  resolve  naturally  to  the  5th  of  the  triad : 


Ex.  653. 


.  Weber. 


GOODRICH  S    ANALYTICAL    HARMONY. 


2/5 


The  notes  b-flat  and  d  occur  in  but  one  other  concord  (G-minor\  but 
since  the  dominant  yth  chord  to  B-flat  major  was  heard  in  the  ca- 
dence, and  as  the  discord  was  resolved  naturally,  there  is  no  doubt 
as  to  the  key-impression.  This  explanation  applies  equally  to  the 
minor  mode. 

The  foregoing  may  be  summarized :  The  root  of  a  concord  can 
not  be  omitted ;  the  3d  is  generally  essential,  but  may  occasionally  be 
left  out  under  the  circumstances  mentioned  ;  the  5th  is  least  essential. 

The  3d  or  5th  of  a  dominant  yth  chord  can  be  omitted,  according 
to  the  nature  of  the  antecedent  harmony  : 


Ex.  654. 


The  3d  is  left  out  at  (a)  because  b-flat  effects  the  modulation,  and 
this  renders  the  leading  note  unnecessary.  At  (b)  the  5th  is  omitted, 
to  prevent  parallel  fifths. 

The  same  is  true  of  the  diminished  yth  chord  when  used  as  a 
transition  harmony : 


Ex.  655. 


The  result  upon  the  tonic  chord  is  the  same  in  both  instances.  (The 
3d  of  a  dominant  yth  is  more  essential  than  the  3d  of  a  diminished 
yth  chord,  because  the  latter  has  no  decided  resolution.) 

Unison  passages,  when  skillfully  constructed,  frequently  suggest 
the  full  harmony  so  plainly  that  no  further  harmonic  accompaniment 
is  required.  Such  an  instance  may  be  found  in  the  cantata  St.  John, 
by  ].  C.  D.  Parker,  from  the  last  measure  of  page  38  to  the  end  of 
page  39,  vocal  score.  The  unison  passage  by  the  chorus  conveys  the 
same  harmonic  impression  as  that  of  the  chord  accompaniment,  and 
it  is  this  circumstance  that  renders  the  unisons  so  effective. 

See  also  Soldiers'  Chorus  from  Faust. 


276 


GOODRICH'S  ANALYTICAL  HARMONY. 


Chapter  LVI. 


RELATED  AND  UNRELATED  TONES. 

o.  HARMONIC  TONE. 

ITT'VERY  tone  that  naturally  belongs  to,  or  forms  part  of,  any  fun- 
-*W  damental  harmony  will  here  be  called  a  Harmonic  Tone.  All 
others  are  unrelated. 

To  indicate  the  related  and  unrelated  notes  in  exercise  work,  the 
author  has  been  in  the  habit  of  marking  them  in  regular  order,  o,  i, 

2,  3»  4.  5.  6- 

A  melody  consisting  entirely  of  harmonic  notes  would,  therefore,, 
be  marked  like  this : 

0000       000 
-3- 


Ex.  656. 


i       i  r 


etc. 


-* *—?- 

I        II 


^2- 


Every  melody  note  here  forms  part  of  the  accompanying  chord. 
The  student  is  familiar  with  this  simple  mode  of  harmonization. 


i.  PASSING  TONE. 


A  passing  tone  occurs  upon  the  unaccented  part  of  a  measure, 
and  does  not  belong  to  the  accompanying  harmony.  In  this  sense 
passing  tones  are  not  harmonized,  as  the  harmony  proceeds  without 
regard  to  their  existence. 


000 


Suppose  an  original  motive  to  be  this,    Ex.  657. 


and  that  it  is  to  be  varied  in  the  following  manner : 

01010 


Ex.  658. 


GOODRICH  S    ANALYTICAL    HARMONY. 


277 


The  intermediate  note  d  is  introduced  in  passing  from  c  up  to  e,  and 
from  e  down  to  c.  The  harmony  remains  the  same,  and  the  passing 
note  is  not,  in  strict  designation,  harmonized : 


01010 


Ex.  659. 


The  original  motive  still  appears,  as  shown  by  the  ciphers.     The 
passing  note  may  ascend,  or  descend,  a  whole  or  a  half  step. 

A  note  of  passage  may  be  introduced  between  the  intervals  of 
any  chord,  but  there  is  no  other  unrelated  note  that  requires  so  much 
care  in  its  treatment.  Being  unrelated,  there  is  danger  of  false  rela- 
tion or  awkward  progression  in  the  two  chords  between  which  it 
occurs.  The  first  example  illustrates  this.  The  ordinary  harmo- 
nization of  the  skip  would  be  to  change  the  position  of  the  C  chord, 


thus :      Ex.  660. 


But  if  the  passing  notes  were 


included  a  very  confused  progression  would  result 


Ex.  661. 


In  such  arrangements  the  proper  test  is  to  repeat  the  chord  with  the 
melody : 


Ex.  662. 


This  reveals  the  faults  more  plainly,  though  the  example  at  (b)  is 
similar  in  effect.     Ex.  659  shows  how  such  errors  are  avoided. 
Another  simple  design  is  this : 


00 


Ex.  663. 


278 


GOODRICH'S  ANALYTICAL  HARMONY. 


Passing  notes  may  be  written  around  this  monotonic  motive  without 
interfering  with  the  harmony  : 


Ex.  664. 


-» •-»*•-] 

3==f=I=^=j 

iT-^-t- 


C|f «*— 

^EtE 


Where  the  passing  notes  ascend  through  the  intervals  of  a  chord  it  is 
better  to  retain  the  harmony  in  the  same  position  : 


Ex.  665. 


01010101        01030101 


The  melodic  notes  that  fall  upon  the  accented  parts  of  the  measure 
form  the  chord  of  B-flat  ;  therefore  this  harmony  constitutes  the 
accompaniment.  Every  other  note,  being  unaccented,  is  a  passing 
note. 

The  melody  notes  of  Ex.  664  may  be  marked  as  harmonic,  and 
arranged  accordingly.: 

00000 

Ex-  666-      T  r  '  = 


The  student  is  to  harmonize  this,  after  which  it  should  be  compared 
with  the  original.     *     *     * 

2.  APPOGGIATURA. 

This  is  almost  the  same  in  effect  as  a  short  suspension,  but  with- 
out preparation.  The  appoggiatura  occurs  on  the  accented  part  of 
a  measure,  and  is  foreign  to  the  accompanying  chord.  The  terra 
appoggiatura.  (meaning  to  lean  ^lpori)  is  here  applied  to  measured 
notes,  as  well  as  to  the  small  note  that  borrows  its  value  from  the 
consequent  measured  note  —  its  resolution.  The  harmonic  appog- 
giatura resolves  up  or  down,  a  major  or  a  minor  2d,  to  some  note 
belonging  to  the  accompanying  chord,  thus  : 


GOODRICH'S  ANALYTICAL  HARMONY. 


279 


2020 


Ex.  667. 


Perhaps  the  scheme  will  appear  simpler  if  the  chord  is  written  in 
notes  of  equal  value  and  a  short  appoggiatura  placed  before  each 
melody  note : 

X 

i — H IV 

Ex.  668. 


Every  pianist  understands  that  the  small  note,  d,  represents  a  tone 
that  is  sounded  with  e  and  g  below,  and  that  c  (the  resolution)  conies 
after.  In  like  manner  the  small  note  /  comes  with  g  and  c,  the  e 
being  sounded  immediately  afterwards. 

The  same  theory  is  here  applied  to  measured  notes. 

The  accompanying  chord  is  to  be  determined  by  the  harmonic 
note  which  follows  the  appoggiatura.  It  is  similar  in  treatment  to 
the  suspension. 

The  student  may  attempt  to  harmonize  the  following  melodic 
phrase  according  to  present  information : 


2020 


020 


This  requires  but  two  chords,  in  their  different  close  positions.  The 
upper  accompaniment  is  to  be  written  in  quarter  notes ;  the  base 
may  consist  of  a  dotted  half  note  in  each  measure.  *  *  * 

Appoggiaturas  may  occur  against  the  different  tones  of  a  discord, 
as  well  as  of  a  concord  : 


Ex.  670. 


The  note  to  which  the  appoggiatura  resolves  is  to  be  omitted  from 
the  accompaniment,  excepting  when  the  melodic  note  is  a  duplicated 
root.  (In  connection  with  this,  see  Suspension,  Ex.  584.) 


280 


GOODRICH'S  ANALYTICAL  HARMONY. 


If  the  chord  progressions  are  correct  the  appoggiaturas  will  not 
interfere  with  the  harmony. 

The  following  authentic  cadence  is  given  for  harmonization  ac- 
cording  to  the  symbols  : 

20202      0 
Ex.  671. 

Examine  the  harmonic  notes  marked  with  ciphers,  in  order  to  deter 
mine  upon  the  chords  to  be  used.  The  harmony  is  then  to  be  a p 
plied  as  though  only  this  outline  appeared : 


Ex.  672. 


In  copying  the  theme  turn  the  stems  upward,  and  wriU-  the  accom 
paniment  in  notes  of  double  the  value  of  the  appoggiaturas.     Thfe 
two  middle  parts  may  follow  the  theme  through  the  different  posi- 
tions of  each  chord.     *     *     * 

The  appoggiatura  can  appear  below,  as  well  as  above  the  har- 
monic note.  In  either  instance  it  is  an  unprepared  dissonance  occur- 
ring on  the  accented  part  of  a  measure.  The  ascending  appoggiatura 
is  usually  written  a  minor  2d  below  the  harmonic  note.  When  the 
appoggiatura  resolves  down  it  merely  follows  the  natural  order  of  the 

scale  : 

2020       2020  2020       2020 


Ex.  673. 


~  "      ~  ~         ~ 


TI^ 


At  (a)  each  resolution  ascends  a  minor  2d.  At  (b)  whole  and  half 
steps  are  used  according  to  the  signature. 

The  same  is  true  of  discords.  An  appoggiatura  may  be  placed 
above  or  below  any  note  of  any  chord.  The  number  of  harmonic 
combinations  that  result  is  almost  incalculable!  There  are  six  in 
the  last  example,  though  only  one  fundamental  harmony  is  used. 

However  easily  the  theorist  may  account  for  these  combinations 
(marked  2),  the  fact  remains  that  a  dissonant  chord  is  heard  on  the 
first  half  of  each  measure  : 


GOODRICH  S   ANALYTICAL    HARMONY. 


281 


Ex.  674. 


We  have,  therefore,  the  benefit  of  this  great  variety,  even  though  in 
the  majority  of  instances  a  trained  ear  will  comprehend  the  nature 
and  resolution  of  these  dissonances,  and  consequently  anticipate  the 
consonance  before  it  is  fully  developed. 

If  the  full  chord  is  sounded  as  an  accompaniment,  while  the  mel- 
ody appears  isolated  and  independent,  every  appoggiatura  (whether 
above  or  below)  will  cause  a  dissonance  during  its  continuance : 


2020        20 


2020 


f                       N*             ^ 

j*  -  |x  :  ^  •-   •    »  ,  r  . 

fr^\          •*" 

Aero  in  i 

cy^S 

J      /o'                      sz                   <S                     /o- 

•    ^             1  &              &             \  & 

—  .j 

^                                                 1 

1 

Ex.  675. 


This  is  founded  upon  a  consonant  triad  as  fundamental  harmony; 
therefore  every  note  that  falls  on  an  accent,  and  does  not  belong 
to  the  accompanying  chord  of  G,  becomes  a  dissonance.  Such  an 
arrangement  presupposes  that  the  solo  is  isolated  from  the  accom- 
paniment. But  where  the  harmony  and  melody  are  connected  by 
means  of  their  proximity  it  is  not  well  to  include  the  harmonic  note 
in  the  middle  parts.  Measures  (a)  and  (b)  illustrate  this : 


Ex.  676. 


The  first  is  much  more  distinct  and  harmonious  than  the  second, 
which  generally  sounds  awkward  and  confusing.  The 'arrangement 
at  (a)  presents  the  additional  advantage  of  two  dissimilar  combina- 
tions, in  place  of  one  : 

o          n 
Ex.  677.  *~ 


The  appoggiatura,  thus  employed,  not  only  unlocks  important  secrets 
of  harmonization,  but  it  enables  the  student  to  greatly  enlarge  his 
repertory  of  tonal  combinations.  *  *  * 


282 


GOODRICH'S  ANALYTICAL  HARMONY. 


The  appoggiatura  may  be  accompanied  with  another  appoggiatura 
according  to  the  same  principles.  This  is  illustrated  by  the  following 
quotation : 


Ex.  678. 


//.  K.  S/ielUy. 


J 

The  b  and  d%  constitute  a  double  appoggiatura  resolving  to  a  and  c%. 
This  is  similar  to  the  double  suspension.  The  following  extract  illus- 
trates in  a  more  elaborate  manner  the  appoggiatura  and  its  treat- 
ment : 


Ex.  679 


I'nn   ll'fber.     Op.  21. 


B:^  —  * 

• 

-&  

0— 

0 

0 

0 

J 

Z.    A  '  '& 

0— 

—0— 

—0— 

—0— 

—  \ 

-* 

^ 

«.. 

—  « 

1 

The  original  sonata  contains  many  other  interesting  illustrations 
this  subject.* 

3.  SUSPENSION. 


of 


No  distinction  is  made  between  the  tied  and  the  untied  suspen- 
sion. Both  are  prepared  in  the  antecedent  harmony,  and  their  treat- 
ment is  similar  to  that  of  the  appoggiatura.  The  suspension  is  less 
bold,  and  where  a  dissonant  combination  is  used  it  palliates  the  harsh- 
ness in  a  manner  not  possible  to  the  appoggiatura. 


-'Transpose  all  the  examples,  and  particularly  observe  numbers  673  and  679. 


GOODRICH'S  ANALYTICAL  HARMONY. 


283 


Chapter  LVII. 


RELATED  AND  UNRELATED  TONES  CONTINUED. 
ILLUSTRATIVE  THEMES. 

4.  ANTICIPATION. 

A  NTICIPATION  occurs  when  a  certain  tone  of  the  consequent 
-"•  chord  is  sounded  in  advance,  usually  on  an  unaccented  part  of 
a  measure,  and  before  the  anticipated  chord  is  heard  in  its  entirety : 


Ex.  680. 


The  first  a,  occurring  on  the  last  of  the  2d  beat,  anticipates  the 
dominant  jth  harmony,  which  falls  upon  the  3d  beat.  The  tonic 
harmony  is  then  anticipated  by  the  short,  melodic  note  g.  These 
bear  some  resemblance  to  passing  notes,  for  both  are  unaccented. 
But  the  passing  note  does  not  belong  to  the  following  harmony. 

The  anticipation  was  much  used  during  the  ijth  century,  particu- 
larly in  the  final  close,  and  it  has,  in  some  strange  way,  become 
known  at  the  "  Haendelian  cadence."  But  this  appellation  is  un- 
justifiable, since  the  anticipation  was  freely  used  by  Italian  compos- 
ers before  Haendel  was  born.  An  instance  is  cited : 


Ex.  681. 


4  0 


From  the  Opera  "Eritrea."     Cavalli  (1600-1670). 
40  40 


2^^^=^j^^ 


The  anticipation  usually  appears 
panied  with  other  notes,  thus  : 


^,  though  it  may  be  accom- 


234 


GOODRICH'S  ANALYTICAL  HARMONY. 


04040 


Ex.  682. 


-  —  i  -  t 

.  V  V  .  -*• 


These  dissonances  (marked  4)  do  not  result  from  suspended  bases, 
because  the  harmony  above  changes  before  the  main  accents.  The 
base  here  is  less  mutable,  and  does  not  quit  its  position  excepting 
upon  the  equal  divisions  of  the  measure.  Compare  this  with  the 
next  example,  in  which  the  dissonances  are  produced  by  means  of 
retardation  in  the  base : 


Ex.  683. 


The  first  movement  to  Beethoven's  sonata,  Op.  31,  No.  i,  contains 
several  instances  of  accompanied  anticipations  similar  in  design  to 
Ex.  682.  See  also  the  dance  of  the  Bayaderes  (No.  II),  from  Rubin- 
stein's Feramors,  after  the  introduction. 

5.     STATIONARY  TONE. 

This  resembles  the  organ-point  in  its  general  features,  but  the 
former  term  applies  to  a  sustained  tone  among  the  upper  parts,  es- 
pecially when  it  is  somewhat  unrelated  to  the  prevailing  harmony. 
Theorists  have  observed  that  the  stationary  tone  does  not  admit  so 
many  dissonant  combinations  as  does  the  low  pedal-note,  and  this  is 
true.  The  former  can  not  be  considered  as  a  ground-work  upon 
which  all  kinds  of  harmonies  may  be  superimposed.  The  following 
quotation  from  a  fugue  will  illustrate  this  : 
Ex.  684. 


GOODRICH  S    ANALYTICAL    HARMONY.  285 

Observe  the  dissonances  in  the  4th  and  5th  measures.  These  would 
not  result  if  the  sustained  note  were  omitted. 

The  object  of  the  stationary  tone  here  (and  in  all  similar  in- 
stances) is  two-fold  :  i.  To  give  consistency  to  the  changing  nature 
of  the  design.  2.  To  fill  in  a  void  between  the  treble  parts  and  the 
base.  These  purposes  justify  the  dissonances,  such  as  e-flat  and  d, 
and  d  and  c-sharp. 

One  of  the  most  remarkable  and  unique  illustrations  of  a  station- 
ary tone  occurs  in  a  song  by  Jensen,  entitled  Marie.  The  last  phrase 
of  the  vocal  part  and  the  postlude  are  quoted  : 


Ex.  685. 


As  an  expression  of  contemplative  admiration  and  constancy  this 
ballad  may  be  classed  among  the  happiest  of  inspirations.  The 
stationary  tone  is  a  prominent  feature  of  the  entire  song,  which  is 
recommended  for  examination  and  analysis. 

The  secret  of  the  student's  success  (if  success  he  wins)  will  con- 
sist in  penetrating  the  design,  and  apprehending  the  object  of  all 
such  devices  as  organ-point,  stationary  tone,  appoggiatura,  suspen- 
sion, etc.,  as  illustrated  by  the  great  creative  artists.  It  is  not  suffi- 
cient to  know  that  a  stationary  note  may  occur  in  any  part  except 
the  real-base,  and  that  dissonant  combinations  thereby  result.  This 
is  superficial  information,  for  the  student  must  also  know  the  com- 
poser's object  in  writing  a  stationary  note.  Why  was  it  included? 
What  would  the  effect  be  were  it  omitted?  Is  it  merely  complemen- 
tal,  or  does  it  add  unity  and  consistency  to  the  design  ? 

6.   EMBELLISHMENT. 

The  author  applies  this  term  to  suspensions  ana  appoggiaturas 
that  do  not  immediately  resolve  to  a  harmonic  tone,  and  also  to  grace 
notes  and  parenthetical  groups.  The  embellishment  is  a  melodic 
license  to  which  the  ordinary  principles  of  harmonization  will  not 
apply.  The  first  example  is  of  an  embellishment  resulting  in  part 
from  suspension : 


286 


GOODRICH'S  ANALYTICAL  HARMONY. 


Ex.  686. 


At  (a)  neither  of  the  first  two  melodic  notes  belong  to  the  accom- 
panying harmony  of  C.  The  same  peculiarity  is  observable  at  (b) 
and  at  (c).  The  suspensions  do  not  resolve  at  once  to  a  consonance, 
but  proceed  to  another  dissonance  before  the  harmonic  interval  ap- 
pears. 

A  similar  instance  is  quoted  from  Beethoven's  Op.  31,  No.  2  : 


Ex.  687. 


1 


m 


The  notes  of  embellishment  are  marked  6.  This  excerpt  contains 
notes  of  anticipation  also. 

The  embellishment  is  an  old  device,  appearing  here  under  a  new 
name  and  with  a  different  explanation.  Numerous  instances  may  be 
found  in  Bach's  Art  of  Fugue. 

This  form  of  embellishment  can  be  reversed  with  equally  good 
effect : 

,—  >--  E.  Xfrin.     Op.  18,  No.  5. 


-» .  ^ 


fc 


i 


The  same  remarks  apply  to  this. 

Sometimes  the  embellishment  includes  three  or  four  notes  foreign 
to  the  accompanying  harmony,  and  at  other  times  a  passing  note  may 
be  included.  Both  styles  are  here  presented : 


GOODRICH  S    ANALYTICAL    HARMONY. 
6 


Ex.  689. 


The  group  of  unrelated  notes  at  (a)  is  in  the  nature  of  a  parenthesis. 
The  embellishment  at  (b)  includes  a  suspension,  harmonic,  and  pass- 
ing note.  Groups  (a)  and  (c)  may  be  considered  as  the  dominant 
harmony  suspended  over  that  of  the  tonic.  These  adventitious  notes 
may  appear  in  a  variety  of  forms,  but  the  four  examples  will  give  the 
student  a  general  knowledge  of  all  others. 

In  transposing  the  exercises  write  the  melody  part  first  with  the 
necessary  ciphers  and  figures  above,  and  then  attempt  the  harmoni- 
zation. There  is  but  little  to  be  gained  from  the  mere  act  of  trans- 
posing an  entire  example. 

The  general  treatment  of  the  embellishment  is  the  same,  whether 
it  begins  with  a  suspension  or  an  appoggiatura.  The  latter  is  neces- 
sarily more  dissonant ;  or,  rather,  the  dissonance  is  unprepared,  and 
consequently  more  noticeable. 

FARTHER  APPLICATION. 

The  artistic  employment  of  these  unrelated  tones  has,  to  a  con- 
siderable extent,  been  illustrated.  In  addition  to  what  has  been 
shown  of  the  general  character  of  passing  tones  it  may  be  remarked 
that  an>-  number  of  them  is  possible,  provided  they  can  be  executed 
between  the  regular  accents  of  a  measure.  Thus,  the  former  design, 


Ex.  690. 


may  be  written  in  i6th  notes  without  interfering  with  the  melodic 
outline  or  the  harmonic  substance  : 


In  both  instances  the  notes  that  fall  upon  the  accented  parts  of  the 
measure  are  c,  e,  g ;    therefore  the  harmony  of  C  will  accompany 


GOODRICIl'S    ANALYTICAL    HARMONY. 


the  melody  in  both  examples.  According  to  a  strict  designation  all 
the  c's  and  ^'s  were  marked  with  ciphers,  as  they  belong  to  the  har- 
mony ;  but  in  actual  practice  all  such  notes  occurring  upon  unac- 
cented parts  of  a  measure  are  passing  notes,  and  will  be  considered 
in  this  light  hereafter. 

Chromatics  may  also  occur  as  notes  of  passage,  two  or  three  being 
included  between  the  harmonic  notes.     A  brief  exercise  is  given  for 

harmonization  : 

o  j  _i o  1  i    ft_iit 011     o 


Observe  that  the  second  and  third  notes  of  each  triplet  are  passing 
notes.     The  ciphers  sufficiently  indicate  the  harmony.     *     *     * 

Chromatics  may  continue  through  a  greater  compass  and  be  har- 
monized collectively,  thus : 


Ex.  693. 


Or,  they  can  be  harmonized  with  the  tonic  chord  throughout.  In 
this  instance  all  the  chromatics  would  be  considered  passing  notes 
with  exception  of  the  first  in  each  measure.  The  harmony  below  is 
a  sufficient  support  for  the  rapidly  passing  chromatics  above. 

A  theme  containing  appoggiaturas,  harmonic  and  passing  notes 
is  here  presented  to  be  harmonized  according  to  the  indications : 


Ex.  694. 


n     i 

0111                 kl       «    t> 

J>1?»S»-  f-»^f- 

•    bi*   «Li    U 

/     7  3 

u.         ^   Jjj   *    ' 

—      1 

ra*/»    •T«    i 

e^^L 

H~Z 

_-      ==-  :t 

t          -«. 

ii~              ' 

t  A  •      ,  •} 

«                                 7 

PH*                » 

)•',.?•» 

r 

i 

-f    :,      t 

^- 

The  notes  unmarked  are  harmonic.     No  modulations  are  suggested 
by  this  theme.     *     *     * 


GOODRICH  S   ANALYTICAL    HARMONY. 


289 


The  following  exercise  contains  the  suspension,  anticipation, 
embellishment,  and  stationary  note.  It  is  to  be  harmonized  accord- 
ingly : 

0    3          0 


Ex.  695. 


The  first  of  this  presents  no  difficulties.  During  the  last  three  meas- 
ures the  harmony  should  make  a  temporary  digression  and  then 
return  to  the  key-note  on  the  last  measure.  In  other  words,  at  the 
beginning  and  ending  of  the  sustained  note  the  chord  is  to  be  that 
of  G-major. 

The  same  exercise  should  be  worked  out  in  Fand  in  E-flat.    *    * 
A  considerable  amount  of  material  is  now  available  in  harmoni- 
zation, and  the  first  point  to  determine  is  the  character  of  the  com- 
position. 

If  a  melodic  figure  like  this — 


Ex.  696. 


should  present  itself,  there  would  be  a  choice  of  several  arrangements : 
i .  The  notes  that  occur  on  the  accented  parts  of  the  measure  may  be 
considered  as  harmonic,  and  the  others  as  passing  notes.  2.  The 
accented  notes  can  be  treated  as  appoggiaturas,  leaving  g,  c,  e  as 
harmonic  notes.  3.  Every  melodic  note  may  be  considered  as  har- 
monic, and  harmonized  separately.  4.  A  series  of  passing  harmo- 
nies may  be  employed  on  a  pedal-note.  The  four  illustrations  are 
presented : 


697 


A     V      1      U      A  P^          «      V 

—     I      i        pi      2000  0  —  ^ — 

-^"•""••SB— fi f      0   TLT E3»=g=*-|:i — i — f  0   Z  t~ 

-^ '-' 1 • L^ 1 i F »-"-^ 1 F-F— 

5^—  i^J 


^f^^ 


•3=^ 


290 


GOODRICH'S  AXALYTICAI.  HARMONY. 


All  these  are  proper,  but  each  has  its  own  peculiarity.  The  first  is 
milder  and  more  smooth  than  the  others ;  (b)  is  the  strongest  and 
most  dissonant ;  (c)  is  rather  detached  and  fragmentary,  and  gives  a 
more  prominent  character  to  each  melodic  tone ;  the  last  arrange- 
ment represents  the  least  possible  disturbance  of  tonic  impression. 

The  nature  of  the  melody,  or  the  situation  in  which  it  occurs 
must,  therefore,  determine  which  will  be  most  suitable. 

Under  this  heading  may  be  mentioned  the  following  exception.*! 
instance  from  the  allegro  of  a  Piano  Quartet : 


Ex.  698. 


A  prepared  suspension  is  here  followed  by  an  anticipation, 
effect  is  somewhat  similar  to  that  of  the  embellishment 


Tne 


GOODkiCH  is   ANALYTICAL    HARMONY.  29] 


PART  XIV. 


Chapter  LVIII. 


HARMONIC  COUNTERPOINT. 

ELEMENTARY  SPECIES. 

SO  much  is  said  and  so  little  is  understood  of  Counterpoint  that 
the  author  has  for  several  years  past  endeavored  to  formulate  a 
system  for  the  simplification  of  this  difficult  subject.     The  results 
are  herein  submitted. 

The  definition  of  Counterpoint  is  note  against  note.  In  its  strict 
sense  this  signifies  two  or  more  voice-parts  moving  independently  of 
one  another,  each  part  having  its  individual  melody.  A  chorus  writ- 
ten in  canonic  or  fugal  style  affords  the  best  illustration  of  counter- 
point. It  presupposes  a  thorough  knowledge  of  all  possible  chord 
formations,  of  inversion,  of  the  theory  of  harmonic  progression,  mod- 
ulation and  transition,  resolution,  harmonization  in  all  its  phases,  sus- 
pension, pedal-point,  chord-relation,  metre,  rhythm,  thematic  devel- 
opment, etc.  Hence  the  term  counterpoint  is  frequently  defined  as 
the  art  of  musical  composition  in  general. 

But  the  word  is  here  used  in  its  specific  sense,  implying  the  union 
of  several  independent  voice-parts  as  in  a  quartet,  and  as  opposed  to 
mere  chord  progression. 

The  author  introduces  what  he  calls  Harmonic  Counterpoint  as 
the  natural  solution  of  strict,  melodic  counterpoint. 

Begin  with  a  simple  design  in  which  notes  of  the  same  value  are 
employed.     The  usual  form  of  writing  the  chords  above,  with 
base  on  a  separate  staff,  will  serve  as  a  nucleus.     The  student  : 
complete  the  following  exercise  : 


29-: 


GOODRICH'S   ANALYTICAL    HARMON' 


Ex.  699. 


Piano  form.     „„„*><***- 

y  i     i        ^5 

-p  —  E^  — 

r     r 

3a_^0—\ 

^^E 

^0      r 

H  1  

i 

-f-  —  i  —  ' 

f_^i  g  ^  — 

"^  ^J  

1      j 

—  g  — 

—  ^  

-^  —  I 

~£  —  ' 

-^  —  i  — 

1 

(2) 


(1) 


It  will  be  a  simple  task  to  arrange  this  in  vocal  form.  In  order  to 
distribute  the  parts  more  equally  with  regard  to  the  intervals  between 
them,  select  the  middle  part  from  each  chord  above  (called  in  former 
lessons  mezzo-soprano],  and  write  this  one  octave  lower  in  the  base 
staff.  The  melody  (soprano),  lowest  treble  part  (contralto),  and  base 
are  to  be  copied  identically.  The  result  is  as  follows : 


Ex.  700. 


This  appears  in  open  position,  and  the  manner  in  which-  it  was  accom- 
plished is  the  very  simplest  method  of  producing  dispersed  harmony. 

NOTE. — When  the  tenor  sings  from  the  treble  staff,  the  notes  sound  one 
octave  lower  than  written.  Therefore  that  part  is  here  represented  exactly  as 
it  is  sung.  Some  books  mention  the  lowest  treble  part  as  tenor,  but  if  that 
part  is  inverted  the  result  will  be  a  very  unequal  distribution  of  the  parts : 


SJ                1                 _                       JJ                 /5                             _" 

-^*^ 

\J         1                     ^^                      ^^ 

'-^ 

"-*• 

»r 

z 

JF    t«                    —  j 

— 

_       1 

M^^          f^ 

^ 

^ 

** 

s> 

\  I/ 

1 

Quartet  form. 

T        -^                              ^;J               ^                   _^_                        ^"            ^.'            -gjj_ 

±  ~ 

P\»                ^               ^ 

<y 

fy 

^        1 

2l;r" 

^  — 

___  

~P  —  ?— 

\  —       

:        \ 

f^             ^^ 

r       1 

\ 

B                                                   [ 

Ex.  701. 


& 


\       I 


J^ 


No  such  void  should  occur  between  the  treble  and  base  parts,  excepting  where 
the  parts  skip  temporarily.  Ex.  700  is  decidedly  better.  This  is  why  the 
author  designated  the  lowest  treble  part  as  contralto,  and  the  middle  one  as 
mezzo-soprano*  Such  designation  presupposes  that  all  the  parts  remain  as 
written,  according  to  former  lessons.  But  when  the  vocal  quartet  is  to  be 
arranged  from  a  design  in  close  position  consider  the  mezzo-soprano  as  tenor, 


'  These  terms  are  applied  to  particular  parts,  even  in  piano  music,  as  a  matter  of  con- 
venience, though  they  have  not  until  the  present  time  been  considered  as  actual  vocal  parts» 


GOODRICH'S    ANALYTICAL    HARMONY. 


293 


and  write  it -an  octave  lower.     This  is  sanctioned  by  musical  custom,  as  may 
be  seen  in  the  following  excerpt  from  the  vocal  score  of  Lohengrin  : 

S  Wagner. 


fix.  702. 


might  -  y      Judge,     I 


Piano. 


*  * 


* 


As  the  quartet  is  unaccompanied  the  piano  part  is  included  merely  for  the 
convenience  of  the  accompanist  at  rehearsal.  Observe  that  the  part  between 
soprano  and  contralto  (marked  T)  is  represented  in  the  accompaniment  an 
octave  lower,  not  according  to  the  notation  of  the  treble  staff.  This  results  in 
an  open  position,  as  it  does  when  sung  by  the  mixed  voices  of  the  quartet. 
Custom  has  made  this  right,  but  to  be  precise  in  its  representation  the  tenor 
part  should  be  written  with  the  tenor  clef. 

This  is  what  the  author  calls  an  elementary  species  of  harmonic 
counterpoint. 

Although  it  is  constructed  by  means  of  chords,  while  strict  coun- 
terpoint is  not,  the  somewhat  independent  appearance  of  the  different 
voice-parts,  together  with  the  fact  that  it  furnishes  a  good  example 
of  plain  quartet  writing,  make  it  very  serviceable  for  present  pur- 
poses. 

As  a  temporary  guide  to  the  student  the  following  limits  for  each 
voice-part  are  given : 

-&- 

Base.    ^^  Tenor.   — —  Contralto.  Soprano,  r?^ 

~&-  —          ^  _ 

F  I'O'         -  '  -      ~ 

°3'EJL = 


294 


GOODRICH'S  ANALYTICAL  HARMONY. 


In  simple  quartet  writing  there  will  be  no  occasion  to  transcend 
these  limits. 

Another  elementary  exercise  is  given  : 


Ex.  70 

*• 

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2: 

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(2) 


(3)     (1) 


Complete  the  simple  harmonization  in  close  position,  and  then  ar- 
range it  in  quartet  form.  Copy  the  first,  third  and  fourth  parts 
(counting  from  the  soprano  downward),  and  write  the  tenor  part  in 
the  base  staff,  one  octave  below  its  original  position.  No  unrulable 
progressions  will  result  from  the  inversion  in  this  example.  Not  all 
designs  admit  this  process  of  inversion,  for  parallel  fourths  in  the 
piano  arrangement  become  parallel  fifths  in  the  open  position.  In 
such  instances  contrary  or  oblique  movement  must  be  employed. 
This  has  been  explained.  *  *  * 

The  last  quartet  arrangement  is  here  presented  for  comparison  : 


This  is  perfectly  correct  and  singable.  There  is  considerable  me- 
lodic character  (to  the  various  voice-parts,  and  a  very  agreeable 
interval  is  maintained  between  soprano  and  contralto,  contralto  and 
tenor,  and  tenor  and  base. 

Elementary  species  of  harmonic  counterpoint  may  contain  any 
natural  chord  progressions,  modulations,  different  kind  of  cadences, 
principal  and  secondary  jth  chords,  skips,  etc.  Unrelated  notes, 
organ-point,  and  chromatic  progressions  are  to  be  reserved  for  the 
other  species. 

A  more  transitional  exercise  is  now  presented  for  arrangement : 


GOODRICH'S    .\XALYTICAL    HARMONY. 


295 


Ex.  706. 


The  inverted  bases  are  an  important  feature  here,  as  they  give  to 
the  lowest  part  a  melodic  movement  very  desirable  in  the  quartet 
form.  In  changing  this  from  close  to  dispersed  harmony  the  stu- 
dent is  to  understand  that  the  tenor  is  the  only  part  to  be  inverted. 

The  tenor  and  the  base  may  occasionally  come  together  upon 
the  unison,  but  the  base  must  not  ascend  above  the  tenor.*  In  such 
instances  write  the  base  an  octave  lower,  for  it  is  not  well  in  these 
exercises  to  alter  the  tenor  part.  *  *  * 

*  The  theme  next  introduced  is  to  be  arranged  in  close  position, 
and  then  as  a  quartet  in  the  manner  explained.  Every  melodic  note 
is  to  be  treated  as  a  harmonic  note.  The  skips  of  a  3d  and  a  4th, 
and  the  temporary  modulations  are  to  be  managed  according  to  the 
usual  methods  : 


Ex.  707. 


-t— i— r 


m 


When  there  is  no  accompaniment  it  is  usually  better  to  so  arrange 
the  essential  discord  that  the  last  tonic  chord  will  appear  in  its  en- 
tirety. The  3d  or  5th  of  the  discord  must  necessarily  be  omitted, 
unless  the  base  is  inverted.  See  Exs.  700  and  706.  *  *  * 

The  modulation  indicated  by  f-sharp  can  be  written  in  various 
ways  : 


*  Some  charming  effects  have  been  produced  by  the  crossing  of  voices,  above  and  below. 
Witness  the  quartet  for  men's  voices  in  Mendelssohn's  42d  Psalm.  But  this  belongs  to  strict 
counterpoint,  andean  not  be  considered  here. 


296 


GOODRICH  S   ANALYTICAL    HARMONY. 


In  the  first  example  the  cadence  is  direct ;  but  this  renders  the  A- 
minor  chord  necessary  on  account  of  the  skip.  At  (b)  an  avoided 
cadence  is  effected.  This  makes  the  F  chord  necessary,  because  of 
the  second  skip,  e  down  to  c.  The  F  chord  is  then  changed  to  that 
of  D-minor  to  avoid  the  awkward  progression  from  sub-dominant  to 
dominant,  when  all  the  parts  ascend,  thus : 


111 


Ex.  709. 


In  the  last  arrangement  (c)  the  5th  resolution  of  the  dominant  yth 
is  used  because  all  the  notes  of  the  fifth  measure  comprise  the  chord 
of  C-major. 

All  these  arrangements  are  available,  and  should  be  used  in  the 
different  transpositions. 

The  same  process  is  to  be  worked  out  in  B-flat  and  in  D-flat. 


Chapter  LIX. 


HARMONIC  COUNTERPOINT  CONTINUED. 


COMPOUND  SPECIES. 


TD  Y  including  in  the  quartet  various  kinds  of  unrelated  notes,  thus 
-L'  making  the  parts  unequal  in  rhythmical  value  and  different  in 
.harmonic  character,  we  produce  compound,  or  mixed  harmonic  coun- 
terpoint. An  avoided  cadence  is  selected  for  preliminary  illustration : 


1*1:1110. 


Ex.  710. 


GOODRICH'S  ANALYTICAL  HARMONY. 


297 


Measures  (a)  and  (b)  are  sufficiently  correct  as  mere  chord  progres- 
sions ;  but  in  a  vocal  quartet  the  effects  are  awkward  and  ragged. 
The  fourths  at  (a)  become  fifths  at  (c)  between  contralto  and  tenor. 
At  (dj  the  parallel  fourths  between  soprano  and  tenor  are  almost  as 
inharmonious  as  parallel  fifths.  Several  methods  for  avoiding  these 
inaccuracies  in  vocal  part-music  are  represented.  The  first  relates 
to  the  upward  resolution  of  the  3d  of  the  discord  : 


Ex.  711. 


The  fourths  and  fifths  are  here  resolved  contrarily,  and  the  example 
thereby  assumes  more  of  a  contrapuntal  character ;  for  this  is  one  of 
the  governing  principles  in  polyphonic  writing.  Not  that  the  coun- 
terpoint shall  consist  of  thirds  and  sixths,  like  a  Glover  duet,  but 
that  these  intervals  represent  the  maximum  of  euphonious  harmony 
between  the  parts,  and  when  so  written  they  can  not  be  wrong. 
Observe  particularly  the  parts  that  ascend  and  those  that  descend. 
The  awkward  progressions  in  Ex.  710,  (c)  and  (d),  may  be 
avoided  also  by  suspending  the  resolution  of  the  3d  and  5th,  or 
of  the  7th,  thus : 


Ex.  712. 


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The  augmented,  followed  by  the  normal  4th,  is  obviated  at  (a) ;  par- 
allel fifths  between  contralto  and  tenor  are  avoided  by  the  suspension 
of  the  7th  at  (b). 

An  important  feature  of  counterpoint,  and  one  that  did  not 
enter  very  prominently  into  previous  lessons,  is  the  separate,  inde- 
pendent movement  of  the  voice-parts.  This  is  partially  illustrated 
in  Exs.  711  and  712.  A  farther  view  is  presented  by  the  follow- 
ing from  Haendel: 


298 


GOODRICH  S     \N.\LYTICAL    HARMONY. 


Ex.  713. 


The  treble  part  is  composed  exclusively  of  harmonic  chord  figures 
in  sequence  order  :  A,  D,  G,  C.  Against  these  the  base  parts  alter- 
nately ascend  alphabetically,  which  necessarily  results  in  dissonating 
intervals.  These  are  marked  2,  for  in  strict  designation  they  are 
appoggiaturas.  This  is  Counterpoint;  therefore  observe  particularly 
the  dissonances,  and  how  these  result.  Perform  and  transpose  the 
last  example.  *  *  * 

Another  important  secret  of  counterpoint  is  to  be  found  in  the 
theory  of  Suspension.  That  is,  any  dissonance  is  allowable  that 
results  from  the  detention  of  one  part  while  another  part  moves 
away  from  (or  toward)  the  stationary  note  : 


Two  \  o  i« 


EX.  714- 


The  second  voice  enters  upon  the  unison  at  (a)  and  descends  a  minor 
2d  at  (b)  while  the  first  voice  sustains  g.  This  results  in  an  extremely 
dissonant  interval.  At  (c)  the  second  voice  descends  another  half 
step,  and  the  dissonance  is  reduced  to  a  minimum.  At  (d)  the  lower 
part  descends  to  e,  and  this  results  in  a  most  agreeable  consonance. 
The  stationary  note  is  its  own  justification,  while  the  lower  part 
moving  away  from  the  suspended  note  presents  a  sufficient  object  for 
the  progressions  described.  A  base  may  be  added  to  this  duet  by 
writing  a  part  in  contrary  movement  to  the  under  treble  part : 


Ex.  715. 


f  he  upper  progression  points  to  the  subdominant ;  therefore  the  base 
melody  leading  up  to  that  note  is  desirable,  especially  as  its  notes 


GOODRICH  S    ANALYTICAL    HARMONY.  299 

harmonize  perfectly  with  those  of  the  contralto.  The  student  may 
form  a  quartet  out  of  this  design  by  adding  another  part  above  the 
stationary  note.  *  *  * 

The  example,  with  its  analysis,  is  presented : 


Ex.  716. 

4^.    j*.  \t-ex-    .&- 


The  base  and  soprano  parts  move  up  at  the  interval  of  a  loth;  the 
tenor  descends  chromatically  in  harmony  with  the  base ;  the  contralto 
remains  stationary.* 

The  student  is  advised  (if  this  design  is  well  understood)  to  write 
a  stationary  note  on  the  tonic  of  the  F  scale,  and,  without  consulting 
the  printed  examples,  add  the  other  parts  synthetically,  so  that  they 
will  correspond  to  the  last  exercise.  Like  all  contrapuntal  designs, 
this  admits  of  re-arrangement  and  inversion,  which  should  now  be 
done.  The -suspended  note  may  appear  in  the  soprano,  contralto, 
or  tenor  parts ;  or  the  base  may  exchange  parts  with  the  soprano. 

#     *     * 

Before  proceeding  with  the  exercises  the  author  will  attempt  to 
give  the  intervals,  considered  separately,  that  produce  the  most  satis- 
factory results  in  two-part  counterpoint.  No  distinction  is  here  made 
between  dissonances  and  consonances,  as  the  object  is  to  show  what 
intervals  may  be  employed  simultaneously,  one  being  a  counterpoint 
to  the  other. 

The  unison  and  octave  come  first.  They  perform  important  parts, 
as  showrn  by  Ex.  716.  Then  the  minor,  major,  and  augmented  2d ; 
minor  and  major  3d ;  imperfect  and  augmented  4th ;  imperfect  and 
augmented  5th ;  minor,  major  and  augmented  6th ;  diminished,  mi- 
nor, and  major  yth.  (The  9th,  loth,  nth  and  i2th  are  here  considered 
as  inversions  of  the  2d,  3d,  4th  and  5th.)  Following  are  these  inter- 
vals in  notation : 


*  A  phrase  of  similar  construction  may  be  found  on  the  first  page  of  Beethoven's  Op.  27, 
No.  i.     It  is  preceded  by  the  reiterated  chord  of  C-major. 


3oo 


GOODRICH'S  ANALYTICAL  HARMONY. 


Ex.  717- 


Some  surprise  may  be  occasioned  by  the  fact  that  the  normal  4th  and 
normal  5th  are  excluded  from  this  scheme,  especially  since  the  5th 
was  employed  so  prominently  by  the  old  contrapuntists.  This  inter- 
val used  to  be  so  highly  esteemed  that  it  was  frequently  included  in 
the  final  chord  to  the  exclusion  of  the  minor  3d  !  Even  during  the 
1  8th  century  certain  composers  preferred  resolving  the  leading  note 
down  a  major  3d  rather  than  omit  the  5th  from  the  final  chord.  How- 
ever, the  4th  and  the  5th  are  generally  unsatisfactory  when  employed 
alone. 

Musicians  of  different  epochs  have  entertained  different  notions 
regarding  the  character  of  intervals.  Until  the  advent  of  Mozart  it 
was  customary  to  terminate  minor  compositions  with  a  major  chord, 
because  the  small  3d  was  not  believed  to  be  sufficiently  euphonious 
for  a  final  ending.  But  Mozart,  whose  delicacy  of  tone-perception 
was  phenomenal,  did  not  accept  all  the  old  canons  and  theories. 
Among  the  important  improvements  that  he  effected  was  the  less 
prominent  treatment  of  the  5th,  and  a  freer  use  of  the  small  3d.  The 
normal  4th  is  even  less  satisfactory  than  the  5th. 

The  situations  in  which  the  5th  may  be  used  were  explained  in 
Chapter  LV  ;  and  when  those  conditions  exist  the  normal  4th  may 
also  be  employed,  one  being  an  inversion  of  the  other  : 


Reinecke. 


Ex.  718. 


In  the  first  measure  the  5th  is  founded  on  the  dominant,  and  th& 
key  is  sufficiently  established  to  determine  the  fact  that  the  major  3d 
(c-sharp)  is  here  omitted.  In  the  second  measure  all  the  intervals 
appear  inverted. 

Where  these  peculiar  conditions  do  not  exist,  the  4th  and  the  5th 


GOODRICH  S    ANALYTICAL    HARMONY. 


301 


will  generally  sound  unsatisfactory.     The  following  instances  are 
cited  in  proof  of  this  : 


Ex.  719. 


-!£ 4- 


The  counterpoint  is  good,  with  exception  of  the  4th  at  (a)  and  thev 
5th  at  (b).  No  reputable  composer  would  write  such  examples  as 
these  in  a  serious  work,  for  they  are  ambiguous  and  unsatisfactory. 
To  preserve  the  original  theme  the  counterpoint  should  be  like 
this  : 

r-  fl-i  ,T  -  1  -  1  —  i  --  i  -  1  --  1 


An  imperfect  5th  takes  the  place  of  the  normal  4th,  and  every  inter- 
val is  satisfactory. 

To  preserve  the  theme  at  (b)  the  counterpoint  is  arranged  in  this 
manner  : 


Ex.  721. 


The  substitute  here  for  the  normal  5th  is  an  augmented  4th.    These 
are  decided  improvements  upon  Ex.  719. 

In  full  harmony  the  4th  and  5th  are  perfectly  proper,  because 
other  intervals  are  combined  with  them.     Thus,  g  in  the  C  chord  is 


not  alone  a  5th  from  c,  but  a  3d  from  e :      Ex.  722. 


ft 


And 


in  the  next  example  the  blank  in  the  upper  parts  is  filled  by  the 
inverted  base : 


Ex.  723. 


No  ambiguity  results  in  such  instances. 

The  principle  of  agreement  between  twro  prominent  voice-parts 


302 


GOODRICH'S  ANALYTICAL  HARMONY. 


(as  illustrated  in  Exs.  720  and  721)  is  frequently  employed  in  full  har- 
mony.    The  following  extract  from  a  Barcarolle  is  a  good  example : 


Ex.  724. 


Tschaiko^^>sky. 


The  real-bases  \vere  especially  designed  to  harmonize  with  the  mel- 
ody, those  two  parts  being  heard  simultaneously  on  the  first  and  last 
of  each  measure.  Note  particularly  the  euphonious  effect  of  the 
intervals  marked  + .  This  is,  of  course,  more  noticeable  when  the 
melody  and  base  are  heard  without  other  harmon}'. 

Hundreds  of  similar  instances  might  be  quoted.     Observe,  for 
instance,  the  relation  between  melody  and  base  in  Schubert's  little 
ballad,  Hedge  Roses,  a  section  of  which  is  quoted : 
Ex.  725. 


The  designs  for  harmonic  counterpoint  are  now  resumed.  Sus- 
pensions and  appoggiaturas  contribute  materially  to  the  independence 
of  the  parts,  and  are,  therefore,  valuable  adjuncts.  F-^r  instance : 


Ex.  726. 


GOODRICH'S  ANALYTICAL  HARMONY 


303 


By  arranging  this  in  open  position  a  fair  specimen  of  the  quartet 
style  is  obtained  : 


Ex.  727. 


Every  musician  perceives  at  a  glance  that  this  is  correct. 

The  first  interval  of  a  7th  becomes  a  6th  ;  the  second  7th  becomes 
a  5th,  and  a  in  the  tenor  part  connects  the  two  chords.  From  this 
point  the  intervals  between  the  middle  parts  are  a  6th,  7th,  and  6th : 


Ex.  728. 


m 


The  main  point  to  observe  is  the  intervals  that  should  not  follow 
each  other  in  similar  movement,  such  as  seconds,  fifths,  sevenths, 
and  octaves.  In  the  last  example  the  7th  resolves  to  a  6th ;  the  5th 
between  contralto  and  soprano  becomes  an  augmented  4th  (/and  6), 
and  this  resolves  to  the  final  6th. 

\Ve  have  here  the  principal  movements :    Oblique  and  contrary.* 
Attention  is  also  directed  to  the  different  rhythmical  denomina- 
tions of  the  notes  in  Ex.  727. 

The  next  exercise  is  to  be  harmonized  in  close  position,  and  after- 
wards arranged  in  quartet  form : 

2020202030203          0 


729. 


V 


p-fe 


The  first  chord  is  designed  to  be  that  of  tonic  major.  C%  in  the  sec- 
ond measure  can  be  treated  as  an  appoggiatura,  or  as  a  harmonic  note. 
On  the  last  half  of  the  third  measure  the  harmony  should  be  changed. 

The  manner  of  producing  the  quartet  remains  the  same :  invert  the 
mezzo-soprano  part  an  octave  lower  and  it  becomes  tenor.  *  *  * 

The  next  theme  contains  passing  notes,  inversions,  modulation, 
suspension,  and  an  anticipation  at  the  close : 


"Students  should  analyze  aii  the  voice-progrressioms  carefully  until  such  analyses 
unnecessary. 


GOODRICH'S  ANALYTICAL  HARMONY. 

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On  the  last  of  the  second  measure  an  8th  resolution  of  the  dominant 
7th  chord  is  outlined ;  or  c  in  the  base  might,  if  the  movement  were 
quick,  be  considered  a  passing  note. 

Complete  the  arrangement  in  piano  form,  and  then  change  it  into 
dispersed  harmony.  *  *  * 

As  the  final  cadence  is  peculiar,  the  student  may  find  these  illus- 
trations useful ;  or  he  may  employ  one  of  his  own  invention  : 


Ex.  731. 


One  more  theme  is  included.     In  the  cadence  a  changing  harmony 
is  to  be  introduced  on  the  resolution  of  an  appoggiatura : 

Ex.  732. 

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It       *                          1 

A*           £? 

»—  *P  —  I-P  —  iz^  

-»  —  "  —  jg-  1 

^i9  

P  —  I  —  4-p  —  P  —  ^  — 

»  —  &  1 

•7 

1         ! 
b 

1 

C|^ 

1  

-1 

^/  ~  —  ^  — 

-*>  *  1 

I—* 


Notes  not  marked  are  harmonic.     Transpose  to  E-flat  and  G,  and 
arrange  in  quartet  form. 


*  These  notes  may  also  be  treated  as  appojjjfiaturas. 
t'i'he  notes  numbered  3  are  treated  as  appoggiaturas. 


ANALYTICAL,  HARMON  v. 


Chapter  LX, 


HARMONIC  COUNTERPOINT  CONCLUDED. 

ORNATE  SPECIES. 

A  STILL  more  independent  and  flowery  species  of  harmonic  coun- 
•£**  terpoint  is  here  introduced.  This  is  accomplished  in  various 
vays.  The  simplest  method  is  to  include  natural  or  chromatic  pass- 
ing notes  in  any  of  the  voice-parts  that  move  a  major  2d  or  a  dia- 
tonic 3d. 

PRELIMINARY   EXAMPLE: 


Ex.  733. 


No  intermediate  passing  note  is  possible  in  the  soprano  part,  nor  in 
the  tenor.  But  between  d  and  e  of  the  contralto  part  a  note  of  pass- 
age can  be  introduced  in  ascending,  and  likewise  in  descending.  As 
these  do  not  interfere  with  the  other  parts  (the  notes  of  the  contralto 
not  being  duplicated)  they  may  be  freely  introduced : 


Ex.  734. 


The  chromatic  alterations  possible  are  too  numerous  to  mention,  but 
a  sufficient  n  imber  will  be  given.  It  is  not  well  to  sharpen  the  root 
above,  unless  the  root  below  remains  as  a  pedal  note : 

-Buck. 


Ex.  735. 


306 


GOODRICH'S  ANALYTICAL  HARMONY. 


Diatonic  passing  notes  may  be  included  in  the  base  whenever  that 
part  proceeds  by  thirds. 

Suppose  this  to  be  the  original : 


Ex.  736. 


m 


Between  the  lower  root-notes  we  may  write  c  and  a,  because  they 
occur  naturally,  and  give  to  the  base  a  regular  melodic  progression : 


Ex.  737. 


Compare  this  with  the  previous  example. 

Of  course  the  passing  tones  must  form  correct  progressions  with 
the  other  parts  and  not  produce  unmusical  relations,  as  in  the  fol- 
lowing : 

zt      I  !- 


Ex.  738. 


In  such  instances  similar  movement  should  be  avoided.  A  simple 
harmonic  design  is  presented  for  elaboration.  First  add  the  other 
three  parts  to  this  base : 


Ex.  739. 


(1)     ' 

After  completing  the  simple  harmonization  as  indicated,  the  passing 
notes  are  to  be  written  in  the  base  between  the  root-notes.     (See  Ex. 


GOODRICH  S    ANALYTICAL    HARMONY. 


307 


737.)     When  any  of  the  upper  parts  move  in  similar  direction  with 
the  base,  care  must  be  exercised  that  no  false  progressions  result. 

•I*      *l*      *!• 

The  chromatic  passing  notes  should  now  be  supplied.  Those 
most  available  here  have  been  illustrated  in  Ex.  734. 

The  passing  note  between  the  3d  and  root  of  the  essential  discord 
may  be  accompanied  with  a  corresponding  note  of  passage  in  the 
soprano  part. 

An  inversion  of  the  design  will  illustrate  this : 


Ex-  740. 


The  root  and  7th  remain  stationary  while  the  other  parts  ascend  and 
descend  in  thirds.  This  may  be  freely  inverted  or  re-arranged.  Such 
designs  are  always  effective,  and  tend  to  relieve  the  monotony  of  com- 
mon chord  progressions.  (See  Ex.  706.) 

When  the  example  is  sufficiently  elaborated  it  is  to  be  arranged 
in  quartet  form  as  usual.  *  *  * 

The  original  base  part  of  Ex.  739  should  be  transposed  into  G, 
and  B-Jlat.  Then  work  it  out  in  the  same  manner.  If  the  base  runs 
too  low  it  is  better  to  skip  up  an  octave  than  a  7th : 


etc. 


Ex.  741. 


The  first  measure  is  not  good  vocally ;  (b)  is  better ;  (c)  is  best  of  all. 
The  second  method  of  producing  this  kind  of  harmonic  counter- 
point is  to  include  more  or  less  ornamentation  in  the  different  parts. 
Unrelated  notes  and  organ-point  will  serve  as  a  nucleus  for  this 
species.  Wherever  the  unrelated  notes  may  appear  they  are  to  be 
treated  in  the  same  manner  as  though  they  occurred  in  the  soprano 
part.  Here,  for  instance,  is  a  design  in  which  a  melodic  figure  passes 
in  sequence-form  through  all  the  voice-parts.  This  would  be  effect- 
ive as  a  vocal  or  an  instrumental  quartet : 


3o8 


GOODRICH'S  ANALYTICAL  HARMONY. 


Ex.  742. 


The  melodic  figure  appears  alternately  in  the  soprano,  contralto,  ten- 
or and  base  parts.  This  relieves  the  effect  of  chord  movement  and 
imparts  to  the  design  a  contrapuntal  character.  The  suspensions 
serve  as  connecting  links,  and  to  prevent  such  awkward  progressions 


as  this,     EX.  743.  : 


which  would  otherwise  have  re- 


sulted between  the  soprano  and  tenor.  Transpose  Ex.  742  into  E- 
flat  and  F. 

Attention  has  been  directed  to  the  fact  that  consecutive  seconds 
should  not  follow  each  other.  The  author  would  also  include  among 
the  unmusical  parallel  movements  two  or  more  normal  fourths,  or  an 
augmented,  followed  by  a  normal  4th.  Parallel  fifths  are  generally 
condemned,  especially  in  counterpoint.  Successive  sevenths  in  simi- 
lar movement  are  as  inharmonious  as  consecutive  seconds.  (It  has 
been  remarked  in  a  previous  chapter  that  augmented  sixths  should 
not  follow'each  other,  for  they  sound  the  same  as  do  minor  sevenths.) 
The  only  intervals  that  remain  for  practical  uses  in  similar  movement 
are :  The  unison,  major  and  minor  thirds ;  major  and  minor  sixths^ 
and  diminished  sevenths.  Unison  passages  (employed  for  the  pur- 
pose of  strengthening  a  certain  melodic  part)  may  occur  in  an}'  twa 
parts,  excepting  the  base  and  soprano.  These  must  not  be  confused 
with  what  are  called  "  parallel  octaves." 

Normal  fourths,  when  they  form  part  of  a  triad  progression,  may 
follow  one  another ;  but  as  they  form  fifths  when  inverted  they  must 
be  used  with  discretion.  In  good  counterpoint  they  are  seldom  em- 
ployed. 

Imperfect  fifths  and  augmented  fourths  can  be  used  in  parallel 
movements  for  a  chromatic  harmonization,  but  these  intervals  must 
be  accompanied  with  some  other  tone  of  a  principal  discord,  as  shown 
in  Chapter  XXXIX.  See  the  coda  to  Au  Matin,  by  Godard. 

So  much  for  the  parallel  progression  of  any  two  voice-parts. 


GOODRICH  S    ANALYTICAL    HARMONY.  309 

Other  parts,  opposed  to  these,  result  in  all  kinds  of  intervals.     But 
these  occur  singly,  not  consecutively,  thus : 


Ex.  744. 


Here  are  seen  a  gth,  a  7th,  and  an  imperfect  5th,  from  the  lowest  to 
the  highest  part.  But  none  of  these  intervals  are  consecutive,  since 
the  two  parts  above  move  in  contrary  direction  to  the  lower  parts. 
The  gth  becomes  a  yth,  the  jth  becomes  a  5th,  and  this  resolves  to 
a  3d. 

This  is  more  contrapuntal  than  harmonic,  and  where  the  upper 
and  lower  parts  move  in  opposite  directions  they  frequently  result 
in  dissonant  combinations  so  harsh  and  incongruous  as  to  be  other- 
wise intolerable.  For  instance,  here  is  a  passage  frequently  used : 


Ex.  745- 


?    *£    *«.  , 


?    +^ 


It  would  be  absurd  to  call  this  harmonic  progression.  Of  the  four 
dissonant  combinations  marked  +  only  the  last  one  resolves  accord- 
ing to  any  rule  or  principle.  The  others  forcibly  pursue  their  way 
to  the  final  tonic  chord.  The  example  consists  merely  of  the  major 
scale  in  contrary  movement,  with  an  additional  counterpoint  added 
in  thirds  and  sixths.  It  is  sufficient  that  the  last  two  chords  har- 
monize. 

The  student  should  analyze  minutely  all  these  illustrative  frag- 
ments in  order  to  discover  the  various  conditions  that  create  the  dis- 
sonant intervals,  thus : 


Ex.  746. 


310 


GOODRICH  S    ANALYTICAL    HARMONY. 


1 

1        1          J-^  1 

f          0 

*      -• 

j-  —  S 

fcP        4 

fl 

0" 

532        p 

I 

S3 

5th, 

1          1 

6th,    9th,  10th,  llth,  10th. 

f~\*     i,        m 

•     f 

^Hj 

'-  —  0  —  g  — 

^1  

V 

This  begins  with  a  consonant  chord.  By  merely  moving  the  solo 
base  to  the  notes  below  and  above  the  tonic,  while  the  upper  parts 
remain  passive,  a  series  of  dissonating  intervals  result.  The  melodic 
groups  in  the  base  revolve  around  the  harmonic  tone  in  a  perfectly 
natural  manner,  and  the  resulting  dissonances  are  not  only  justifi- 
able, but  desirable. 

The   next   illustration  is  somewhat  similar,  though  there  is  a 
counter-theme  above : 


Ex.  747. 


o     i     o     i     a     o 

The  fractional  figures  in  the  middle  show  the  intervals  from  the  base 
to  the  soprano ;  the  other  figures  indicate  the  character  of  the  unre- 
lated notes.  As  the  two  middle  parts  remain  passive,  merely  consider 
the  extreme  parts.  The  yt.h  is  a  passing  note,  and  the  Qth  results 
from  the  contrary  movement  of  base  and  soprano.  This  discord  on 
the  last  beat  is  the  dominant  yth,  the  d  below  being  a  passing  appog- 
giatura  that  occurs  naturally  in  the  descending  melodic  figure.  The 
dissonant  gth  becomes  a  consonant  loth  immediately.  At  the  close 
the  solo  base  ends  upon  the  tonic,  while  the  upper  parts  do  not  resolve 
until  later.  Thus  all  the  disagreeing  intervals  are  the  result  of  favor- 
ing circumstances,  and  present  an  unmistakable  object  for  their  ap- 
pearance. 

The  pedal-note  may  also  form  a  nucleus  for  this  ornate  species  ; 
though  the  suspended  base  presents  some  difficulties  to  the  singer, 
especially  if  it  be  long  continued,  or  if  the  upper  harmony  is  of  a 
chromatic  character.  The  well  known  quando  corpus  from  Rossini's 
Stabat  Mater  is  an  instance.  This  was  written  as  an  unaccom- 
panied quartet,  but  the  author  has  known  professional  singers  to 
fail  miserably  in  their  attempts  upon  this  composition.  The  fol- 
lowing extract  from  an  original  Hymn  is  somewhat  similar,  but 
the  chorus  bases  are  here  re-inforced  by  the  'cellos,  double  bases, 
horns,  and  kettle-drum: 


CrOODRICHS    ANALYTICAL    HARMONY. 
ChoniB  and  Orchestra. 


3" 


Ex.  748. 


f          O  -  ra    pro      no  -  bis,       no  -  bis   pec  -  ca       to  -  ri  -  bus 

i 

£ 


;     m — i— 


Tim*. 


-IfP 2      J.      +      V     '» * 


II 


^^ ~ *      I     |  |  g 

20— *  rrjr  j.  ^ 


-*— — ^3— - 


^t= 


nunc      et     in     ho  -  ra    mor  -  tis,        in     ho  -  ra  mor  -  tis      nos    -    trae 


P^ 


» 


This  begins  with  the  tonic  chord,  and  while  the  bases  remain  as 
pedal-note  the  other  parts  ascend  chromatically  through  a  series  of 
major  chords.  In  the  6th  measure  the  base  becomes  again  consonant 
to  the  other  parts.  Such  designs  are  impossible  to  the  average  chorus 
singer,  unless  assistance  is  afforded  by  the  instrumental  accompani- 
ment. 

Pedal  passages  like  the  following  present  no  difficulties  and  are 
always  effective : 


Ex.  749. 


The  dissonances  are  well  prepared  and  of  brief  duration.  The  upper 
parts  are  sufficiently  independent  to  form  good  counterpoint  with  one 
another  against  the  pedal-note. 

The  embellishment  and  the  stationary  tone  can  be  made  to  serve 
good  purposes  in  florid  harmonic  counterpoint.  These  may  occur 
in  any  of  the  upper  parts,  though  the  former  is  better  suited  to  the 
soprano  part  if  it  contains  several  foreign  tones.  A  simple  illustra- 
tion of  this  follows : 


312 


GOODRICH'S  ANALYTICAL  HARMONY. 


Mnrttti — t-rm — I — 

||||||=|| 


Ex.  750. 


This  is  especially  adapted  to  the  quartet  form,  for  if  the  tenor  part 
were  written  an  octave  higher  the  embellishments  would  sound  some- 
what confused.  As  it  is,  the  distance  between  the  voice-parts  is  both 
convenient  and  agreeable. 

Finally  a  series  of  suspensions  may  be  introduced  with  the  same 
general  result.  The  principle  of  suspension  as  explained  in  a  previ- 
ous chapter  is  not  difficult  of  comprehension,  but  some  of  its  phases 
are  so  complex  as  to  require  both  skill  and  ingenuity  in  their  man- 
agement. A  familiar  design  is  selected  as  a  nucleus : 


Ex.  751. 


Suspend  the  progression  of  the  lower  part  from  the  last  half  of  each 
measure,  thus : 


Ex.  752 


•i 


«i 
F=f^ 


The  elementary  theory  is  illustrated  here.  This  can  be  utilized  in 
various  ways  :  A  third  part  may  be  added  a  6th  below  the  soprano 
and  the  sequence  thus  continued. 

An  additional  advantage  to  this  plan  is  that  it  can  be  freely 
inverted,  and  the  retardations  may  occur  in  the  soprano  or  tenor 
parts.  The  manner  in  which  the  chords  follow  one  another  does 
not  admit  a  series  of  fundamental  bases ;  but  the  design  naturally 
rests  upon  a  tonic  pedal-note.  An  example  of  this,  with  the  suspen- 
'sions  in  the  tenor  part,  is  represented : 


Ex.  753. 


w*  ^ 

52  

•C' 

—  j^  



Ej  —  j—  I 

«y 

A' 

^    J- 

at"" 

1  19  ' 

° 

—  —  ^—  H 

r  r 

•^ 

^    tt    [^  — 

—  £  , 

__  _^  , 

r—  

—  ^  _J 

GOODRICH'S    ANALYTICAL    HARMONY. 


313 


The  tenor  part  becomes  more  prominent  on  account  of  the  suspen- 
sions, and  a  very  ordinary  progression, 


is  thus  made  interesting. 

The  student  should  re-arrange  Ex.  753  with  the  suspensions  in 
the  soprano  part.  The  descending  tendency  of  the  progressions 
must  be  considered,  that  the  tenor  may  not  interfere  with  the  pedal- 
note.  Therefore  it  will  be  necessary  to  begin  the  tenor  part  at  a 
considerable  interval  from  the  base,  as  at  (a),  unless  the  former  be 
made  more  florid,  as  at  (b)  : 


Ex.  755. 


In  an  instrumental  quartet  the  second  arrangement  would  be  pref- 
erable. 

Or  the  rhythm  could  be  enlivened  by  brief  imitative  passages 
between  the  middle  parts,  thus  : 


Ex.  756. 


It  would  be  a  useful  practice  to  transpose  the  base  and  soprano  of 
this  into  A-flat  and  G,  and  then  attempt  to  supply  the  middle  parts 
without  consulting  the  printed  exercise. 

This  chapter  will  be  concluded  with  a  quotation  from  a  choral 
melody  harmonized  by  that  Past  Master  of  counterpoint,  Sebastian 
Bach: 


GOODRICH'S  ANALYTICAL  HARMONY. 


The  passing  notes  and  suspensions  are  here  indicated  that  the  reader 
may  more  readily  appreciate  the  construction.  Aside  from  these  un- 
related notes  appearing  in  the  various  voice-parts,  this  is  based  upon 
a  simple  harmonic  design  not  materially  different  from  some  of  the 
previous  illustrations.  In  fact,  the  aim  has  been  to  deduce  principles 
from  the  works  of  standard  composers  and  to  lead  gradually  to  this 
point  in  quartet  writing,  for  the  practice  of  all  masters  is  similar 
in  these  respects.  The  underlying  principles  are  fundamental,  and 
must  be  the  same  in  all  countries. 

We  know  that  certain  passing  notes  may  be  filled  in  between 
certain  intervals  of  any  harmonic  design  ;  and  that  appoggiaturas, 
suspensions,  anticipations,  and  stationary  tones  can  be  included  in 
any  part,  either  for  the  sake  of  grace  and  ornamentation,  or  to  make 
a  particular  voice-part  more  independent.  Added  to  this  information 
we  know  what  single  intervals  produce  the  most  satisfying  effects 
when  heard  simultaneously,  and  what  intervals  may  safely  follow 
each  other  consecutively.  Thus  equipped,  the  intelligent  student 
will  experience  no  great  difficulty  in  comprehending  the  intricacies 
of  canon  and  fugue. 

The  examples  of  the  different  species  of  harmonic  counterpoint 
may  serve  for  organ  arrangements  in  dispersed  harmony,  for  a  simple 
string  quartet,  or  for  a  vocal  quartet  or  chorus. 


GOODRICH'S  ANALYTICAL  HARMONY. 


Chapter  LXI. 


HARMONIC  ACCOMPANIMENT  ILLUSTRATED. 

THE  accompaniment  is  now  to  be  considered  separately.     In  a 
general  sense  it  is  secondary  to  the  melody,  though  the  best 
accompaniments  are  those  that  are  complete  in  themselves. 

Primarily  the  accompaniment  is  to  consist  of  certain  chords  sug- 
gested by  the  melody,  and  these  chords  are  to  be  arranged  according 
to  the  principles  of  harmonic  progression.  Observe  the  following 
section  of  a  theme  from  Rubinstein's  Op.  13  : 


Ex.  758. 


The  harmonies  are  sufficiently  indicated  here.  In  the  second  meas- 
ure c%  is,  of  course,  an  appoggiatura.  A  simple  arrangement  of  this 
may  be  attempted.  *  *  * 

The  composer's  solution  is  given,  not  for  comparison,  but  as  a 
study : 


Ex.  759. 


•      f -«..  v — — i       ^ 

*=r     =^±±1^=*=: 
— ^~~~ r^ —       ,  .    ^ 


Any  position  of  a  chord  may  be  used  in  the  accompaniment,  this 
being  a  matter  of  taste,  rather  than  of  theory.  A  certain  position 
might,  however,  bring  the  accompaniment'  into  such  proximity  to 
the  melody  that  the  effect  would  be  indistinct  or  confused. 

In  the  next  fragment  the  melody  traverses  a  considerable  space, 
and  in  such  instances  the  accompaniment,  being  very  nearly  station- 
ary, appears  first  below  and  then  above  the  theme  * 


316 


GOODRICH 'S   ANALYTICAL    HARMONY 


Ex.  760. 


Where  the  melody  does  not  skip,  the  chords  may  be  re-arranged 
in  this  manner : 


Ex.  761. 


This  form  makes  the  accessory  parts  more  prominent,  and  is  bettet 
suited  to  a  sustained  melody  in  measured  notes. 

The  broken  chord,  or  arpeggio  form,  is  much  used  in  accompani- 
ments. It  is  usually  advisable  to  conduct  the  chords  as  though  they 
occurred  in  regular  progression.  For  instance,  the  design  at  (a),  if 
performed  in  sixteenths,  would  appear  as  at  (b)  in  this  example : 


-tf-T1?  r—j- 

—  —  «  

\kr-*-t  E 

U*  :-^-l 

•/       .5.          • 

f\*    li      <5"     * 

^  * 

° 

Ex.  762. 


3: 


l=C=Fg 


(2)  (2) 


(2) 


(2) 


This  is  not  always  essential,  but  it  is  always  correct.  Until  the  stu- 
dent has  acquired  some  experience  in  this  matter  it  would  be  well  to 
perform  such  designs  as  the  last  (b)  in  simultaneous  form,  as  at  (a), 
in  order  to  test  the  correctness  of  the  chord  progressions,  for  they 
are  in  a  harmonic  sense  identical. 

From  this  last  we  may  derive  the  harp-like  form,  embracing  two 
or  more  octaves : 


GOODRICH  S    ANALYTICAL    HARMONY. 


3-7 


Ex.  763- 


ff  i  i  i  r  f    fc —  *  f  ?=  -  i  !  *  ^^ 

^*^^        •»-—  !      i      * 


^r-^u 


This  arpeggio  style  is  conducted  in  the  same  manner,  only  the  latter 
is  more  ornamental. 

A  design  like  this  (or  like  that  of  the  previous  example)  once 
begun  should  be  continued,  at  least  during  the  length  of  a  period. 
The  accompaniment  to  the  soprano  solo  Inflammatus  represents  a 
more  independent  type.  It  consists  of  passing  appoggiaturas  above 
and  below  the  reiterated  chords.  One  phrase  of  this  will  suffice  to 
show  the  design  : 


Ex.  764. 


This  style  of  accompaniment  is  continued  throughout  the  solo  part. 

It  is  not  the  purpose  here  to  give  a  great  variety  of  styles  for 
these  adventitious  parts,  but  merely  to  indicate  their  character.  The 
broken  chord  or  arpeggio  forms  represent  the  same  harmonic  sub- 
stance, and  are  generally  conducted  in  the  same  manner  as  chord  pro- 
gressions. But  there  are  some  additional  features  to  be  mentioned. 

i.  In  relation  to  melodic  skips  that  could  not  very  well  be  har- 
monized individually  on  account  of  their  length.  Such  is  the  fol- 
lowing : 


GOODRICH'S  ANALYTICAL  HARMONY. 


Ex.  765. 


]*••*••*•  •*-   » 

|-T£fel    =^qg     ,^>  :      — ^-u^^g^     =p=q 
\-^^^^         =f=  =^  =^ 


To  attempt  to  follow  these  melodic  skips  with  the  chord  accompani- 
ment would  be  impracticable,  even  if  it  were  desirable.  As  here 
written  the  piano  part  is  simple,  correct  and  effective. 

2.  Where  the  theme  is  so  rapid  as  to  make  it  impossible  to  move 
the  harmony  at  the  same  rate  of  speed.  In  such  instances  merely 
mark  the  rhythm  and  indicate  the  harmonic  substance,  thus : 


Ex.  766. 


The  harmonic  impression  created  by  the  theme  is  fully  represented 
by  the  accompaniment,  and  .on.  account  of  the  florid  nature  of  the 
violin  part  the  accompaniment  is  made  as  simple  as  possible.  This 
principle  may  be  generally  applied. 

3.  Still  another  situation  presents  itself  when  the  melody  pro- 
gresses in  such  manner  as  to  violate  the  rules  of  re-solution  if  the 
harmony  should  follow  the  theme: 


Ex.  767. 


GOODRICH  S    ANALYTICAL    HARMONY. 


The  skip  from  the  leading-note  down  a  major  jth  to  the  tonic  would 
certainly  be  incorrect  as  a  harmonic  progression,  but  as  the  accom- 
paniment is  conducted  regularly  the  melody  may  skip  about  as  the 
fancy  of  the  composer  suggests.  A  somewhat  similar  instance  occurs 
in  that  excellent  song  "  O  wretched  slave,"  from  Paul  and  Virginia: 


a 

f                 Mass's. 
L<2.  

Ex.  768.   / 

1 

jf    ff  f  ^  5—  • 



vT)         ^         *            * 

4 

our  trust    be-trayed 

n 

f                                        ^                                Itf                            M^ 

0      0      *      J 

Ku        5       ^       ^ 

-•  e-*-«  — 

^ 

—  *  0      0  0  ' 

t-wj  —  •  —  •  —  0  — 

—  &  1 

S                                 MM 

1 

The  le?p  of  a  gth  in  the  vocal  part  is  characteristic  of  the  senti- 
ment, and  was  a  clever  stroke,  but  this  does  not  apply  to  the  har- 
mony. Observe  that  the  chords  in  the  accompaniment  as  written 
by  the  composer  are  in  strict  conformity  to  our  principles  of  har- 
monic progression. 

In  reference  to  appoggiaturas  and  suspensions  the  author  has  had 
occasion  to  remark  that  the  resolution  of  the  dissonant  tone  is  omitted 
from  the  remaining  parts  of  the  harmony  whenever  the  melody  and 
accompaniment  are  comprehended  in  one  design,  as  here : 


Ex.  769. 


u    'i     o       a o 


s^          r^ — IT 
=F^i=i=^-*-^\ 

9  ^^0          t^~~  ^ 

•  "*""  * 


Either  the  harmonic  quartet,  or  such  piano  arrangements  as  this,  are 
to  be  treated  in  the  manner  illustrated.  Each  of  the  harmonic  notes 
indicated  by  ciphers  (when  preceded  by  an  appoggiatura)  is  to  be 
omitted  from  the  accompanying  harmony, — the  root  in  the  base 
always  excepted.  But  if  the  melody  issued  from  a  different  instru- 
ment, and  especially  in  a  higher  or  lower  register  so  as  to  separate  it 
from  the  accompaniment,  then  it  would  not  be  necessary  to  omit  nnv 
part  of  the  harmony  on  account  of  the  temporary  dissonances.  Com- 
pare this  with  the  previous  example  : 


GOODRICH'S  ANALYTICAL  HARMONY. 


a     o       2-^0 

T  •- 


Ex.  770. 


The  fact  that  the  solo  is  here  considerably  removed  from  the  piano- 
part  renders  this  plan  more  necessary ;  and  it  is  also  better  to  make 
the  accompaniment  complete  in  itself — especially  when  the  solo 
issues  from  a  different  kind  of  an  instrument,  as  in  the  last  example. 
Arrangers  frequently  make  the  mistake  of  writing  blank  and  naked 
intervals  in  the  piano  part,  trusting  that  the  other  instrument  will 
complete  the  effect.  This  is  generally  a  false  hope,  especially  if  the 
timbre  of  the  two  instruments  is  different,  as  in  this  instance  from  a 
Rigaudon  by  Rameau : 


Ex.  771. 


The  great  French  composer  and  theorist  is  not  responsible  for  the 
unsatisfactory  thinness  of  the  accompaniment,  as  this  is  an  arrange- 
ment for  violin  and  piano  by  W.  L,enz.  If  both  parts  were  played 
by  the  pianist  the  fault  would  disappear,  but  where  a  violin  or  flute 
plays  the  melody  the  piano  part  sounds  unsatisfactory. 

In  the  following  cadence  to  a  piano  duo  by  Bach  this  point  is 
farther  illustrated.  A  blank  5th  occurs  in  the  first  piano  part,  but 
this  vacuum  is  supplied  by  the  second  piano,  which  sounds  the 
major  3d: 


GOODRICH'S  ANALYTICAL  HARMONY. 


32* 


Ex.  772. 


„        _    f   •       •- 

=*= 

.  » 


l-iuno  II. 


The  effect  is  the  same  as  though  the  last  chord  issued  from  one 
instrument,  excepting  the  unison  D.  Both  D-minor  fugues  are  writ- 
ten in  this  manner,  and  no  fault  appears. 

Another  form  of  accompaniment  consists  in  duplicating  the  un- 
related notes  of  the  melody  either  in  the  unison  or  octave.  In  vocal 
music  this  affords  some  assistance  to  the  singer  (though  this  is  seldom 
necessary),  and  adds  to  the  interest  of  the  associate  part,  thus: 


Ex.  773. 


Voioe. 


Lassen. 


The  principal  occasion  for  this  style  of  accompaniment  presents  itself 
here,  where  the  proximity  of  the  vocal  and  instrumental  parts  renders 
it  necessary.  The  effect  of  the  unrelated  tones  is  more  noticeable 
because  of  their  duplication  in  the  piano  part.  The  additional  disso- 
nances that  would  otherwise  result  are  here  absent.  Even  when  the 
melody  is  somewhat  removed  from  the  associate  parts  this  plan  may 
t>e  adopted  with  good  effect.  Observe  this  illustration : 


OOODRICH'S  ANALYTICAL  HARMONY. 


//.   W  Nicholl. 


Ex.  774. 


^        ^j^b       ^g 

~3.£:   I  :  -^-^rr^g-.  .^  .   l^.:^ 

J.    "i  .  J. 


333 


^^ 
?S 


r-^-r 


Where  the  accompaniment  proceeds  with  the  regular  fundamental 
harmonies,  irrespective  of  the  disagreeing  tones  in  the  solo,  it  pre- 
supposes that  the  melody  is  more  animated  and  ornamental,  and  that 
it  is  considerably  removed  from  the  neighborhood  of  the  accompani- 
ment: 

Oboe. 


Ex.  775. 


Under  the  circumstances  this  is  sufficiently  proper ;  but  the  fact  re- 
mains that  this  plan  is  generally  inferior  to  that  of  Exs.  773  and  774. 

The  most  common  form  of  accompaniment  to  a  quartet  or  chorus 
is  to  transcribe  the  vocal  parts  almost  exactly.  When  these  are  inde- 
pendent they  neither  require  nor  admit  the  same  amount  of  extrane- 
ous embellishment  as  does  a  solo.  Even  to  some  of  the  examples 
of  harmonic  counterpoint  a  figufated  accompaniment  would  sound 
incongruous  and  confused.  (Such,  for  instance,  as  numbers  742  and 
749.)  If  the  vocal  parts  are  in  plain  harmony  we  may  repeat  the 
chords  in  notes  of  quicker  succession,  or  employ  the  arpeggio  or 
broken  chord  forms. 

Or  a  design  more  or  less  ornamental  may  be  given  the  instru- 
mental parts,  as  in  the  tournament  of  Song  from  Tannhauser ;  the 
chorus  "  Down  with  the  Moslem  ! "  from  Buck's  Don  Munio ;  the 
last  chorus  in  Dream  Pictures  by  G.  E.  Whiting ;  or  the  Kermesse 
in  the  second  act  of  Faust.  In  the  latter  the  principal  themes  of  the 
well-known  waltz  are  heard  in  the  orchestra  as  a  musical  coloring  to 
the  scene,  while  the  chorus  parts  merely  consist  of  harmonic  outlines. 
See  also  the  last  of  No.  3  in  Mendelssohn's  95th  Psalm,  after  the  trip- 
let figures  appear  in  the  accompaniment. 


GOODRICH'S  ANALYTICAL  HARMONY. 


323 


A  brief  quotation   from  Haydn's  Imperial  Mass  will  show  the 
usual  mode  of  procedure  when  the  vocal  parts  are  merely  harmonic : 


Ex.  776. 


Glo    -    ri  -  a,     glo     -     ri  -  a, 


Glo    -    ri  -  a,     glo     -     ri  -  a. 

B    •»•  -ft-      •*- 

^ &        • - — —ZT 


*— *- 


The  instrumental  base  is  slightly  more  animated  than  the  chorus 
base,  and  the  former  is  also  altered  in  a  few  unimportant  particulars. 
When  the  choral  parts  are  more  measured,  buoyancy  and  anima- 
tion may  be  imparted  to  the  movement  by  adopting  some  such  plan 
as  this  from  the  finale  to  "  The  Heavens  are  Telling" : 


The  Creation.'1 


324 


GOODRICH'S  ANALYTICAL,  HARMONY. 


Some  designs  do  not,  on  account  of  their  simplicity  or  their  com- 
pleteness, require  any  elaboration  in  the  accompaniment.  Among 
numerous  instances  of  this  kind  the  following  from  a  four-part  song 
by  Sir  G.  A.  Macfarren  is  selected  : 


Ex.  778. 


l#y       *~~ 

-r  

rj  1 

r 

^=^ 

Sr- 

i* 

Sigh     no  more,    la  -    dies, 

c 

PT^  — 

-H  ^-*- 

—  1  1  — 

\  \/ 

»       • 

^T                  0 

T  = 

^ 

*—  i  — 

?»- 

V 

^^— 

|  

L      _ 

-1  — 

—  9  —  •  "if  f_  

~~P  *  

E5s 

G' 

* 

\\  )y                                   X 

^ 

1              1 

Sigh     no  more,    la  -    dies,    sigh    no  more,    la  -   dies, 

f\* 

, 

a        * 

•  •  j                                  *  * 

^> 

;          * 

b 

_l  

-6  ,  r-*-=  fS  

-i  —  i  —  i  —  i  —  i 

/    L                   —  -— 

^  

- 

«                , 

j 

—  -^  — 

9 

r 

^ 

BE 

^..       » 

*y 

^     JL  • 

••-         •»•      *•          ^ 

r-  r  4 

f        -&-        •+- 

rv« 

•j 

1 

•  • 

eS         « 

J«i 

'            9        \ 

2    > 

^j 

i 

The  accompaniment  here  is  a  literal  copy  of  the  vocal  score,  the 
tenor  part  being  represented  in  the  base  staff  according  to  the  real 
pitch  when  sung  by  a  man.  The  accompaniment  to  part  songs  of 
this  character  may  very  well  be  dispensed  with,  except  for  purposes 
of  rehearsal. 

As  a  continuation  of  this  subject  the  student  will  receive  much 
benefit  from  selecting  simple  songs,  copying  the  melody,  and  then 
writing  an  accompaniment  according  to  the  nature  of  the  theme  and 
the  plans  herein  suggested.  Afterwards  more  elaborate  songs  may 
be  chosen  for  harmonization. 

The  accompaniments  to  a  volume  of  choice  songs  are  in  them- 
selves an  excellent  study,  and  in  many  of  the  modern  classical  songs, 
the  instrumental  part  is  frequently  the  most  important. 


GOODRICH  S   ANALYTICAL    HARMONY, 


PART  XV. 


Chapter  LXII. 


INTERDICTED  PROGRESSIONS  AND 
FALSE  RELATION. 

THE  author  has  thought  best  to  treat  of  these  topics  secondarily, 
as  they  might  occur  in  connection  with  the  various  lessons;  for 
Tie  believes  that  much  of  what  is  commonly  "  forbidden  "  is  mere 
bugbear. 

Mozart  and  Beethoven  were  hampered  by  the  countless  rules  and 
restrictions  of  theorists,  and  the  young  composer  who  is  ambitious 
will  be  obliged  to  defy  many  of  these  injunctions,  or  suppress  his 
originality. 

CONSECUTIVE  PARALLEL  FIFTHS. 

These  have  occasioned  more  discussion  than  any  other  progres- 
sion, and  under  ordinary  circumstances  they  are  certainly  incongru- 
ous and  unsatisfactory.  The  prohibition  of  parallel  fifths  should 
apply  more  particularly  to  the  quartet  style  of  writing,  and  to  places 
wherein  the  fifths  occur  prominently.  But  if  a  composer  desires 
to  produce  a  rough,  rigid,  or  blank  effect,  he  may  purposely  choose 
these  interdicted  intervals,  as  Mr.  E.  S.  Kelley  has  done  in  his  Mac- 
beth music. 

Where  the  fifths  occur  in  the  lower  or  middle  parts,  and  are 
somewhat  concealed  by  the  melody  above,  there  is  little  or  no  reason 
in  condemning  them.  So  many  instances  like  the  following  have 
been  written  that  it  is  scarcely  necessary  to  explain  them  : 


326 


GOODRICH  S   ANALYTICAL    HARMONY. 

Beethcrven. 


Ex.  779. 


5  5 


The  e-flat  being  retained  throughout  the  measure,  and  the /of  the 
melody,  both  contribute  to  the  good  effect,  which  is  really  this : 


Ex.  780. 


Grieg,  in  his  Op.  35,  repeats  a  phrase  a  minor  3d  above,  each  chord 
being  in  its  first  position.    All  the  parts  ascend  in  similar  movement  : 


Ex.  781. 


Had  such  a  passage  come  under  the  notice  of  Fiix,  Kirnberger,  G. 
Weber,  or  Marx,  they  would  have  been  horrified ;  for  not  only  are 
the  parallel  fifths  unconcealed,  but  they  are  emphasized  with  strong 
accents.  And  a  fault  equally  grievous,  in  the  opinion  of  past  theo- 
rists, lies  in  the  "  cross  relation  "  of  both  phrases.  Yet  how  sug- 
gestive of  the  quaintness  and  incitement  of  northern  life  are  these 
very  transgressions  of  musical  rule  ! 

The  next  example  presents  parallel  fifths  moving  by  regular 
steps : 


Grieg.     Op.  22. 


Ex.  782. 


''* 


«V: 


•»     K 


GOODRICH'S  ANALYTICAL  HARMONY. 


327 


The  melody  is  above,  and  the  fifths  occur  in  the  accompaniment. 
These  are  paliating  circumstances,  if  any  be  needed.  But  in  truth 
the  principal  charm  lies  in  the  unusual  mode  of  harmonization,  which 
infuses  into  the  music  a  distinctive  character  and  coloration. 

A  remarkable  passage,  containing  a  series  of  corresponding  fifths, 
is  here  quoted  from  the  ballet  music  in  Rubinstein's  Per  amors : 


Ex.  783. 


"Bride  of  Cashmere" 


m 


These  result  from  the  resolution  of  an  augmented  6th  chord,  No.  r, 
direct  to  a  major  chord.  But  this  is  necessary  on  account  of  the 
chromatic  progressions. 

Theorists  agree  in  allowing  the  imperfect  to  follow  the  normal 
5th ;    the  reverse  of  this  is  not  so  good : 


Ex.  784. 


§15 


The  effect  at  (b)  is  inclined  to  be  rough  and  generally  unsatisfactory. 
By  retaining  the  tonic  a  more  satisfactory  result  is  obtained : 


Paine. 


Ex.  785. 


CV-U ! 


Such  progressions  as  the  following  from  Meyerbeer  (a)  are  of 
frequent  occurrence,  though  this  fact  does  not  excuse  them.  At  (b) 
the  faults  are  avoided,  and  without  changing  the  melody  or  the  har- 
mony: 


32* 


GOODRICH'S  ANALYTICAL  HARMONY. 


Ex.  186. 


"HIDDEN"  FIFTHS  AND  OCTAVES. 

Another  restriction  of  harmony  book-makers,  and  one  that  has 
very  little  practical  justification,  is  in  relation  to  hidden,  or  covered, 
fifths.  They  occur  in  such  progressions  as  these, 


Ex.  787. 


which  have  already  been  explained.  But  these  speculative  gentle- 
men write  a  passing  note  between  e  and  g  to  show  that  a  5th  can 
be  imagined  when  the  contralto  skips  up  a  3d.  The  author  has  no 
hesitancy  in  asserting  that  this  so-called  fault  lies  entirely  in  the 
imagination  of  those  who  consider  it  incorrect;  for  the  supposed 
tone  (/)  has  no  relationship  to  the  C  chord,  and  unless  the  inter- 
mediate tones  were  produced  .by  means  of  portamento  no  fault  could 
occur.  It  is  a  common  progression  to  be  met  with  in  almost  every 
composition — even  in  the  fugues  of  Bach. 

"  Hidden  Octaves  "  constitute  another  theoretical  bugbear.  One 
can  scarcely  conceive  of  a  composition  that  does  not  contain  hidden 
octaves,  and  yet  they  have  been  cataloged  among  the  guilty  things 
to  be  avoided.  Here  is  an  example : 

1 \ 


Ex.  "88. 


I 


i 


9: 


Observe  that  b  in  the  soprano  part  ascends  a  half  step  to  c,  while  the 
base  ascends  a  normal  4th,  from  c  to  c,  thus  producing  an  octave. 
The  other  octave  (s  below)  exists  in  the  imagination,  so  we  are  told, 
and  therefore  it  is  classed  with  the  "  hidden,"  the  "  covered,"  and 
the  "  secret "  fifths  as  something  about  which  a  suspicion  is  attached. 


GOODRICH'S  ANALYTICAL  HARMONY. 


329 


But  these  progressions  are  so  necessary  in  modern  harmony,  and 
they  have  been  written  so  persistently  by  standard  composers,  that 
it  seems  a  mere  waste  of  words  to  discuss  the  matter. 

Of  course  a  distinction  should  be  made  between  chord  movements 
in  free  instrumental  style  and  regular  harmonic  progression  in  strict 
style.  Such  movements  as  the  following  are  frequently  written  in 
piano  and  organ  music  : 


Liszt. 


Ex.  7^9. 


But  in  a  vocal  composition  this  would  be  unmelodious  and  contrary 
to  the  principles  of  harmonic  succession. 

Our  illustrations  show  that  any  of  these  interdicted  progressions 
can  be  used.  It  is  for  the  composer  to  decide  whether  he  desires  the 
peculiar  effect  which  they  produce. 

CROSS   RELATION. 

The  quotation  from  Grieg  illustrates  this : 


Ex.  790. 


The  upper  part  proceeds  from  e  to  g,  while  the  middle  part  moves 
from  c  to  e-flat.  This  is  called  cross  relation.  It  is  more  noticeable 
at  (a)  and  (b)  in  the  next  example : 


Neither  of  these  can  be  recommended,  for  such  anomalies  usually 
leave  an  unpleasant  impression  on  the  mind. 

The  student  must  not  confuse  this  with  a  mere  change  in  mode, 
where  the  chromatic  alteration  occurs  in  the  same  voice-part :  . 


Ex.  792. 


330 


GOODRICH'S  AMAI.VTICAL  HARMONY. 


Such  relations  are  strictly  correct.  E.  F.  Richter  has  very  properly 
pointed  out  that  the  interdiction  ought  to  he  removed  from  all  such 
progressions  as  these : 


Ex:  793. 


Indeed,  composers  have  long  since  settled  this  matter.  The  chorale 
quoted  from  Bach  (Ex.  757)  presents  an  instance  of  false  re1?!'  m 
where  the  B-major  chord  is  succeeded  by  a  dominant  7th  ot?  R  • 


Ex.  794. 


But  the  7th  chord  containing  d-natural  is  not  connected  with  the  & 
•major  chord,  which  latter  occurs  at  the  end  of  a  musical  and  poetical 
phrase. 

Numerous  examples  similar  to  the  following  might  be  quoted  : 


Beethoven. 


Ex.  795. 


The  semi-phrases  here  begin  upon  the  second  quarter  of  each  meas- 
ure. 

THE  AUGMENTED  SECOND. 

The  productive  musicians  have  used  this  so  variously  and  so  effect- 
ively that  all  attempts  to  suppress  it  have  proved  futile.  The  author 
finds  nothing  wrong  in  the  progression  of  an  augmented  2d,  either 
melo'dically  or  harmonically,  and  he  can  not,  therefore,  condemn  it 
Two  illustrations  will  suffice  : 


Ex.  796. 


GOODRICH  S    ANALYTICAL    HARMONY. 


Mendelssohn.  Glinka. 

fc^ESt   =g-HHSg;— *^L     *     I 


^ 


H 


"=Fr=%=F 
=^F *H 


It  /s  simply  a  natural  interval  of  an  important  modern  scale.  The 
reverse  of  this  has  been  considered  still  more  "faulty";  yet  both 
progressions  are  rendered  necessary  by  the  nature  of  the  harmonic 
minor  scale.  (See  chorus  of  the  Shemites  in  Rubinstein's  Tower  of 
Babel.} 

TRITONE. 

Another  peculiar  prohibition  applies  to  the  "  tritone,"  4  to  7  of 
the  major  scale.  It  is  so-called  because  of  the  three  whole  steps 
included  in  this  interval: 


Ex.  797-  Gfc= 


tzac 


The  tritone  has  incurred  the  displeasure  of  theoretical  writers  to  such 
an  extent  that  they  have  actually  objected  to  this  common  progres- 
sion : 


Ex.  798. 


In  order  to  be  "  allowable  "  the  tritone  must,  we  are  told,  belong  to 

the  same  chord : 

"Ttt. 
1=Z=- 


Ex.  799. 


^=^ 


But  this  is  no  better  than  the  preceding  example.  The  difficulty  of 
singing  the  augmented  4th  led,  originally,  to  its  exclusion  from  poly- 
phonic music.  The  prohibition  was  then  generally  applied  to  all 
styles,  and  even  the  relation,  or  the  suggestion,  of  the  tritone  was 
interdicted.  In  truth,  the  majority  of  these  interdictions  belong  to 


33* 


GOODRICH'S  ANALYTICAL  HARMONY. 


the  primitive  stages  of  musical  development.  But  they  have  been 
repeated  in  all  seriousness  by  modern  writers,  as  though  we  had 
made  no  progress  since  Peri  composed  his  Euridice.  They  ought  to 
have  been  discarded  during  the  last  century,  for  very  few  of  these 
prohibitions  have  any  application  to  modern  musical  composition. 
A  certain  interval,  or  chord  progression,  may  sound  unmelodic  or 
inharmonious,  because  the  situation  in  which  it  occurs  is  not  favor- 
able, or  because  no  object  appears  to  make  it  necessary.  This  does 
not  justify  us  in  condemning  the  procedure,  for  the  composer  with  a 
definite  object  in  view  may  produce  excellent  results  with  material 
that  seemed  useless  to  the  mere  speculator.  Thus  Mr.  E.  Prout 
writes  an  example  of  these  minor  triads, 


Ex.  800. 


3^ 


and  because  they  sound  strangely  in  his  ears  he  declares  them  to  be 
"simply  detestable  !  "  But  Mr.  Prout  ignores  the  fact,  as  do  nearly 
all  speculative  musicians,  that  genius  is  a  law  unto  itself,  and  that  a 
Saint-Saens,  Dvorak,  Grieg,  or  Tschaikowski  may  discover  in  the 
strangeness  of  a  certain  progression  the  very  expression  they  desire 
to  convey. 

Indeed,  all  these  "  detestable  "  harmonizations  have  been  utilized 
by  the  greatest  composers,  and  it  is  merest  folly  that  attempts  to 
proscribe  them.  It  may  be  well  for  the  arranger  to  be  restricted  by 
abstract  formulas,  but  the  composer  who  has  something  new  to  say 
through  the  mystic  soul-language  can  not  be  bound  and  fettered  by 
arbitrary,  didactic  theorems  that  must  be  violated  on  every  page  of 
original  music.  While  musical  effects  remain  inexhaustible,  theory 
must  play  a  secondary 


GOODRICH  S    ANALYTICAL,    HARMONY. 


333 


Chapter  LXIli. 


ANALYSIS  OF  HARMONIC  SEQUENCE. 

MELODIC  sequence  is  the  repetition  of  a  group  or  ngure  upon 
different  degrees  of  the  scale ;  the  consequent  succession  of 
similar  melodic  intervals. 

The  author  applies  this  term  to  harmonic  progression  and  transi- 
tion, wherever  a  certain  arrangement  of  chords  is  repeated  higher  or 
lower. 

There  must  be  some  characteristic  feature  to  the  original  progres- 
sion, which  is  considered  as  the  design.  The  position,  or  the  kind 
of  chords  employed,  or  the  manner  in  which  they  follow  one  another, 
must  be  sufficiently  characteristic  to  constitute  a  model.  Here  is  a 
simple  illustration  of  harmonic  sequence  : 


Ex.  801. 


The  model  (a)  is  repeated  exactly  at  (b),  a  whole  step  lower.  In 
each  measure  a  dominant  chord  with  the  3d  uppermost  resolves  to 
its  major  tonic,  and  in  both  instances  the  base  descends  from  the  root 
to  the  minor  7th,  which  latter  resolves  to  the  3d  of  the  major  concord. 
This  is  strict  sequence,  as  is  the  following : 


Ex.  802. 


$t- ;  I  s    *= 

**. .•    -S-  -*• 


^P 


The  design  (a),  consisting  of  the  peculiar  resolution  of  an  essential 
discord,  is  repeated  in  sequence  at  (b)  and  at  (c). 


334 


GOODRICH'S  ANALYTICAL  HARMONY. 


A  free  sequence  mry  be  described  as  a  repetition  in  which  the 
same  positions,  but  not  the  same  species,  of  chords  are  employed. 
There  is  this  important  difference  between  the  two:  Strict  sequence 
is  transitional ;  free  sequence  is  not.  The  latter  is  here  illustrated  : 


Ex.  8or 


•)•• 


=?; 


The  arrangement  of  the  chords  with  reference  to  their  positions  is 
the  same  in  every  measure ;  hence  they  appear  identical  to  the  eye. 
But  at  (a)  the  chords  are  minor,  at  (b)  they  are  major.  There  is  also 
a  passing  7th  on  each  second  beat,  but  some  of  these  are  major,  and 
some  minor  sevenths.  At  (c)  the  last  chord  is  an  imperfect  triad, 
though  this  appears  in  its  first  inversion,  as  do  all  the  others.  No 
transitions  here  occur,  since  the  natural  tones  of  but  one  scale  are 
employed.  Compare  this  with  Exs.  801  and  802.  Another  kind  of 
sequence  takes  place  when  several  chords  follow  one  another  in  the 
same  position.  A  familiar  example  may  be  quoted  from  Beethoven's 
Op.  2,  No.  3,  where  the  first  eleven  chords  appear  in  their  second 
position : 


Ex.  804. 


The  sequence  is  indicated  by  the  slur. 

Two  other  examples  are  quoted  from  the  same  opus : 
Ex.  805. 


fiijxi=r~r~?"<  J— -1 
r^-f-f-ff^  ff— 

^«,      I          ^"    «4          ^*^« 


The  first  is  an  irregular  sequence.     Both  are  in  the  free  style. 

The  melodic  part  of  a  sequence  from  Haendel's  F-major  Chaconne 
is  now  presented  for  the  student  to  complete  according  to  the  model : 


Ex.  806. 


GOODRICH'S  ANALYTICAL  HARMONY. 


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33: 


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The  design  to  be  carried  out  begins  at  (a).     At  (b),  (c)  and  (d)  it  :'s 
to  be  sequenced,  all  three  parts  being  similar  to  those  given  at  (a). 

*    *    * 

The  Chaconne  from  which  this  extract  was  taken  is  constructed 
principally  by  means  of  sequence,  and  would  be  a  useful  study  in 
connection  with  this  subject. 

The  next  illustration  represents  diminished  and  dominant  yth 
chords  resolving  to  the  note  above  the  root  of  each  discord : 


/.  Ldvt. 


Ex.  807. 


The  4th  resolution  of  the  dominant  yth  chord  in  the  second  measure 
corresponds  to  the  first  resolution  of  the  diminished  chord  in  the 
other  measures.  That  is,  the  base  in  every  instance  ascends  a  minor 
2d  from  root  to  root.  This  is  free  sequence. 

A  charming  chromatic  sequence  is  contained  in  the  following 
excerpt : 


Chopin.    Op.  6,  No.  1. 


#*        '•  — ^t_        **m L^_|^^^fc         |  >—v| 


The  harmonic  design  consists  of  diminished,  changed  to  correspond- 
ing dominant  yth,  chords.     It  is  all  strict  with  exception  of  the  last 


336 


GOODRICH'S  ANALYTICAL  HARMONY. 


melodic  note.  Here  ihe/%  is  used  in  place  of/  5,  because  the  return. 
of  the  first  theme  requires  the  former.  Chopin  was  partial  to  these 
sequence  designs,  and  used  them  most  adroitly.  It  would  therefore 
be  well  to  transpose  the  last  example,  and  to  seek  other  illustrations 
in  his  works. 

Exs.  809  (a)  and  (b)  are  to  be  completed  by  the  student.  The 
first  consists  in  resolving  a  series  of  essential  discords  indirectly. 
The  figures  indicate  3d  and  4th  resolutions. 

The  design  of  Ex.  809  (b)  should  be  analyzed,  and  then  con- 
tinued to  a  natural  cadence  on  D.  No  chromatic  signs  are  here 
necessary. 


Ex.  8oga. 


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Transpose,  but  do  not  invert,  these  exercises. 


K3ODRICH  S    ANALYTICAL    HARMONY. 


337 


Chapter  LXIV. 


INFLUENCE  OF   RHYTHM   AND  PHRASING  UPON 
HARMONIC  MOVEMENT. 


WE  enter  here  into  a  new  field,  and  one  that  will  require  some 
knowledge  of  musical  analysis.  Chord  movements  that  are 
contrary  to  all  principles  of  harmonic  progression  frequently  occur 
in  seeming  connection  with  one  another.  To  analyze  the  design  is 
particularly  necessary  in  such  instances.  We  must  know  the  con- 
structional divisions  of  the  work  in  order  to  understand  where  the 
harmonic  connection  ceases. 

Periods,  sections,  and  sometimes  even  phrases,  are  to  be  isolated 
from  what  follows,  and  the  principles  of  progression  and  resolution 
do  not  necessarily  apply  beyond  these  divisional  or  subdivisional 
points. 

Antiphonal  groups,  sequences,  echoes  have  a  like  effect  upon  the 
harmonic  connection,  as  well  as  upon  the  phrasing.  An  illustrative 
instance  is  quoted  from  a  descriptive  song.  Notice  particularly  the 
progression  from  the  second  to  the  third  measure : 


Ex.  810. 


Mililotti. 


•&— •- 


1 


r 


The  first  two  measures  comprise  a  phrase.  At  (c)  a  different  senti- 
ment is  represented.  This  is  indicated  by  the  composer :  "  Like  a 
boat-song  heard  in  the  distance."  There  is  therefore  no  connection 
between  the  phrase  ending  at  (b)  and  the  one  commencing  at  (c). 
If  there  were,  the  resolution  of  the  essential  discord  at  ^D)  would 
certainly  be  unrulable,  and  even  incorrect.  The  next  quotation  is 
of  similar  import : 


338 


GOODRICH'S  ANALYTICAL  HAK.MON 


Schumann. 


Ex.  811. 


^ipEEjEj 
^  .-sS^-—s- 


The  phrases  marked  p  and  /  were  intended  by  the  composer  to  be 
isolated.  Everything  indicates  this  fact.  Consequently  the  pro- 
gression from  the  A  to  the  G  chord  does  not  come  within  the  rule 
or  meaning  of  a  continuous  harmonic  movement ;  they  are  disunited, 
both  in  phrase  and  sentiment. 

The  next  illustration  is  taken  from  a  mazourka  by  Chopin.     A 
part  of  the  prelude  and  one  phrase  of  the  principal  theme  are  given : 


Ex.  812. 


M=MfH-i 


The  mazourka  begins  at  (b),  after  the  preliminary  motive,  with  which 
the  piece  closes.  No  rest  or  pause  appears  between  the  two  phrases 
(a)  and  (b),  but  the  design  is  so  apparent  that  the  composer  merely 
wrote  two  perpendicular  bars  to  indicate  the  beginning  of  the  ma- 
zourka. Otherwise  the  progression  from  the  second  to  the  third 
measures  would  be  unaccountably  and  inexcusably  strange. 

A  motive  or  phrase  repeated  in  form  of  an  echo  is  to  be  included 
among  these  seeming  contradictions  to  the  principles  of  harmonic 
movement.  Such  an  instance  is  cited  from  Beethoven's  Op.  27, 
No.  i: 


Ex.  813. 


GOODRICH'S  ANALYTICAL  HARMONY. 


339 


The  short  figure  at  (a)  is  repeated  an  octave  higher  at  (b).  The 
groups  are  not  only  separated  by  the  slurs,  but  the  tone-quality  is 
considerably  changed ;  for  these  echoes  are  given  to  different  kinds 
of  instruments  in  orchestral  music.  As  Beethoven  generally  had  an 
eye  to  the  orchestra  while  composing,  the  design  may  be  represented 
in  this  manner : 


Ex.  814. 


String*. 


-3r- 
tf-b- 


Jf-af- 


m 


All  such  examples  are  to  be  understood  in  this  sense. 

As  the  period  is  a  point  of  repose  we  may  very  properly  conclude 
that  chord  connection  does  not  extend  beyond  this  point,  unless  the 
directions  of  the  composer  indicate  the  contrary. 

United  periods  are  usually  bound  together  and  treated  as  one 
period.  A  rest,  or  pause,  or  the  addition  of  other  parts  above  or 
below  the  prevailing  harmony,  are  sufficient  to  suspend  the  rules 
of  progression  at  that  particular  point.  All  these  circumstances  are 
to  be  duly  considered  by  the  student. 

Mention  must  now  be  made  of  an  important  consideration  that 
enters  here,  and  one  that  has  hitherto  been  overlooked.  It  is  the 
license  frequently  made  necessary  by  the  natural  tendency  of  a  har- 
monic sequence : 


Ex.  815. 


34° 


GOODRICH'S  ANALYTICAL  HARMONY. 


The  model  (a)  is  sequenced  at  (b)  and  (c).  The  progressions  are 
perfectly  correct  in  each  measure  considered  separately.  But  from 
one  measure  to  another  the  chord  movements  are  not  so  good,  as 
may  be  seen  by  changing  the  order  of  the  sequence  : 


Ex.  816. 


These  parallel  movements,  especially  to  an  inverted  base,  lack  poise, 
and  are  unmusical.      But  according  to  the  original,  only  the  half- 


cadence-figure   is  connected,       EX.  817. 


3|== 


the  next 


measure  being  a  sequence  of  this.  Consequently  the  progressions 
at  (a),  (b)  and  (c)  are  all  disconnected. 

Sequence  thus  renders  possible  progressions  of  this  kind  that 
would  otherwise  be  undesirable,  and  even  incorrect. 

A  better  illustration  is  quoted  from  the  finale  to  Beethoven's 
Op,  27,  No.  i  : 


Ex.  818. 


The  second  semi-phrase,  marked  f,  does  not  succeed  the  first 
according  to  any  known  principle  of  chord  progression  ;  for  all  the 
parts  skip  down  a  considerable  distance.  And  if  the  sequence  had 
followed  in  the  same  octave  the  result,  as  a  continuous  progression, 
would  have  been  still  more  irregular : 


aThe  slur  is  to  be  understood  in  its  usual  sense,  as  indicating  the  notes  that  are  to  be 
connected. 

fThe  author  includes  the  slurs  merely  to  show  the  design.    The  style  is  staccato. 


GOODRICH'S    ANALYTICAL    HARMONY.  34! 


Ex.  819. 


Kven  a  less  particular  composer  than  Beethoven  would  avoid  such 
inaccuracies  as  these,  and  yet  the  original  arrangement,  with  the 
downward  skip,  would  be  equally  objectionable  but  for  the  sequence- 
like  character  of  the  antiphonal  groups,  which  are  disconnected  from 
each  other.  In  the  performance  of  such  passages  every  intelligent 
pianist  understands  that  the  different  groups  are  antiphonal,  and 
therefore  he  not  only  separates  one  from  the  other,  but  changes  the 
tone-quality  also. 

There  are  many  forms  of  harmonic  sequence,  but  as  they  all 
possess  the  same  feature  in  common  it  will  be  sufficient  to  present 
one  more  illustration : 


Ex.  820. 


In  each  measure  here  the  chord  passes  into  its  first  inversion,  and 
the  positions  are  the  same  at  (a),  (b)  and  (c).  This  constitutes  the 
sequence.  The  example  likewise  embraces  the  license  previously 
mentioned,  and  but  for  the  sequence  these  progressions  would  be 
unsatisfactory. 


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GOODRICH  S   ANALYTICAL    HARMC'NV. 


PART  XVI. 


Chapter  LXV. 


HARMONY  IN  FIVE,  SIX,  SEVEN,  EIGHT  AND 
TEN  PARTS 

THE  simplest  method  of  producing  harmony  in  five  parts  is  to 
duplicate  the  base  an  octave  below. 
One  quotation  will  suffice : 


•0- 

±      ±      .*-      «•>                  «. 

Chcfiti 

Op.  n. 

L  *:> 

.V  **    X 

5-—  JL^|      i*  *^ 

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Ex.  821. 


This  requires  no  farther  explanation  than  that  of  the  first  sentence. 
The  next  design  is  very  simple : 

Sckumattx. 

=*=r 


Ex.  822. 


y    \     s  \ 


i 


:fc 


The  two  upper  parts  are  duplicated  below,  with  the  addition  of  a  in 
the  middle. 


GOODRICH'S  ANALYTICAL,  HARMONY. 


Another  method  is  represented  in  the  following  : 


Ex.  823. 


343 


j      » 

a 

H 

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— 

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The  main  object  of  the  baritone  part  is  to  fill  in  the  void  between 
base  and  treble  parts.  In  piano  music  this  is  a  comparatively  unim- 
portant office,  but  in  orchestral  scores  these  vacuums  usually  sound 
bare  and  unsatisfactory.  This  form  is  more  compact  than  where  the 
added  part  is  a  mere  duplication  of  the  base.  The  parallel  octaves 
that  may  result  between  the  baritone  and  lower  treble  parts  are  not 
objectionable,  but  the  general  principles  of  harmonic  progression 
should  be  observed.  The  best  tones  for  duplication  are  roots  and 
fifths.  Designs  similar  to  the  last  require  the  most  care  in  their 
management,  because  the  parts  are  independent  of  each  other. 

SIX  AND  SEVEN  PARTS. 

By  writing  four  treble  parts  and  adding  an  octave  base,  six-part 
harmony  is  produced.     A  comparatively  new  feature  enters  here : 


Ex.  824. 


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Octaves  between  the  extreme  treble  parts  are  unobjectionable,  be- 
cause the  lowest  treble  part  merely  reinforces  the  melody.  The 
unisons  in  the  base  have  been  explained. 

The  base  should  move  contrarily  as  much  as  possible,  though 
when  there  is  a  connecting  note  above,  the  base  may  move  in  similar 
direction  with  the  treble  parts  to  prevent  the  former  from  descending 
too  low.  See  second  measure  of  Ex.  824.  When  there  is  no  con- 
necting link,  contrary  movement  is  especial!}7  necessary. 


344  GOODRICH'S  ANALYTICAL  HARMONY. 

The  following  quotation  from  Schumann  is  in  this  style : 


Ex.  825. 


*—  &CJ 3=        —  J*^  |    I       j— 

^**   5  5  55^-*   *   5 


In  order  to  give  the  bases  a  melodic  progression  contrary  to  the 
upper  parts  the  composer  doubles  the  minor  3d  in  three  parts  at 
(a),  and  the  5th  in  four  parts  at  (b),  which  is  rather  weak.  But  at 
(c)  these  inversions  and  duplications  are  justified.  At  (e)  the  com- 
poser leaves  to  implication  the  root  of  the  A-major  chord  in  order  to 
avoid  the  open  fifths : 

-jJFJrnr- 

sr-q^F 


Ex.  826. 


The  perfect  cadence-form  here  enables  us  to  readily  seize  upon  the 
object  in  view.  Otherwise  the  a  could  easily  have  been  supplied  by 
including  it  in  the  previous  chord : 


Ex.  827. 


Another  simple  design  consists  of  two  chord-groups  o\  three  parts 
each.     The  following  form  has  been  much  used : 


Ex.  828. 


-& 


*  —  i—*  -  f  -  m  —  r*  -  1  - 


On  account  of  the  connecting  note  in  each  chord-group  this  is  per- 


GOODRICH  S    ANALYTICAL    HARMONY. 


345 


fectly  simple  in  its  construction.  Observe  that  the  3d  and  jth  of  the 
<iiscord  are  not  duplicated ;  and  that  when  the  3d  or  5th  of  the  con- 
cord is  doubled,  the  duplicated  parts  move  contrarily.  Ex.  828  should 
be  re-arranged. 

By  duplicating  the  melody,  or  the  base,  seven-part  harmony  is 
obtained : 


— *- 0 1 r-0 — j-* 0 9 r-» 


Ex.  829. 


Observe  the  contrary  and  oblique  movements.  At  (b)  the  dominant 
pedal-note  is  preferable  to  a  duplication  of  the  real-base.  These  six 
and  seven-part  designs  frequently  result  in  parallel  octaves,  and  some- 
times in  fifths.  Observe  this  quotation  from  Beethoven : 


Op.  14,  No.  1. 


Er.  830. 


The  lower  group  is  a  duplicate  of  the  upper.  Parallel  octaves  there- 
fore result.  This  duplication  is  a  peculiar  feature  of  the  entire  move- 
ment, and  it  needs  no  approving  sign  from  the  theorist. 

In  the  allegro  to  Op.  31,  No.  2,  Beethoven  wrote  a  still  more  irreg- 
ular progression,  the  result  of  doubling  a  tone-group  : 


Ex.  831. 


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346 


GOODRICH'S  ANALYTICAL  HARMONY. 


Had  the  composer  submitted  this  to  his  old  teacher,  Albrechtsberger, 
it  would  have  been  unhesitatingly  expunged.  But  as  we  listen  to 
the  effect  we  hear  a  simple  design  duplicated  in  the  octave.  This  is 
of  frequent  occurrence  in  modern  music.* 

EIGHT,  NINE,  AND  MORE,  PARTS. 

Since  the  enlargement  of  the  piano  key-board  the  tendency  has 
been  toward  heavy  masses  of  chords  and  extended  harmonies. 
A  moderate  example  is  quoted  from  Bendel : 


Ex.  832. 


--"-- 


r    r 
1     '  " 


C»ran<lioHO. 


3=5± 


As  the  octave  base  is  continued  in  vibration  by  means  of  the  damper 
pedal  the  harmony  may  be  said  to  contain  ten  parts. 

Here  may  be  mentioned  the  first  of  the  solo  part  in  Liszt's  E-flat 
concerto,  with  its  ponderous  chord-effects,  written  in  open  defiance 
of  all  the  thorough-base  formulas. 

Eight  and  ten  simultaneous  parts  frequently  occur  in  organ  scores, 
for  two  of  these  can  be  given  by  the  pedals.  Herewith  is  a  specimen 
from  Lemmen's  Christmas  Offertory  : 

Ex.  833. 


ff 


m 


*  A  similar  instance  occurs  in  the  little  Romance  that  usually  follows  the  Traumerei  by 
Schumann. 


COODRICH  S   ANALYTICAL    HARMONY. 


-347 


In  such  a  irass  of  harmony,  and  in  the  midst  of  so  much  noise,  it  is 
not  possible  to  maintain  much  clearness  or  purity  of  style. 

According  to  the  principles  herein  set  forth  almost  any  number 
of  harmonic  parts  are  feasible.  Certain  chord  progressions  that  admit 
of  free  nrversion  may  be  duplicated  to  the  utmost  limit : 


Ex.  834. 


J 

j*_*_*iiiM_jfc. 
»— • — S&-S*Tg= 


Designs  similar  to  this  appear  in  almost  every  orchestral  score,  and 
it  could  also  be  utilized  in  an  arrangement  for  two  pianos,  or  piano 
arid  organ,  (c)  and  (d)  are  mere  duplicates  of  (a)  and  (b). 


34* 


GOODRICH 'S    ANALYTICAL    HARMONY. 


Chapter  LXVI. 


ABRUPT,  ENHARMONIC,  AND  REMOTE 
TRANSITION. 

KEY  AND  CHORD  RELATIONS. 

/"COMPOSERS  have  furnished  an  abundance  of  examples  with 
V-'  which  to  illustrate  these  subjects. 

In  addition  to  the  means  of  modulation  previously  explained, 
various  methods  have  been  employed,  and  these  will  be  examined. 

The  present  object  is  to  go  beyond  the  boundaries  of  the  closely 
related  keys.  We  now  recognize  a  connection  between  various 
tonalities  that  were  during  the  time  of  Mozart,  and  even  since  then, 
considered  remote. 

At  first  a  given  tone  is  selected  as  a  means  of  connecting  two 
apparently  unrelated  keys.  If,  for  example,  a  cadence  takes  place 
upon  D  we  may,  by  retaining  that  tone,  pass  directly  to  any  of  the 
following  keys : 


At  (a)  the  same  tonic  remains,  and  the  dominating  chord  is  identical 
in  both  modes.  At  (b)  the  new  tonic  is  situated  a  major  3d  below 
the  old.  Three  tones  of  the  B-flat  scale  occui  in  that  of  D;  four  are 
foreign.  The  D  and  B-flat  are  what  Hauptmann  called  "  parallel 
keys,"  and  their  relationship  is  sufficiently  established.  At  (c)  we 
pass  to  G-minor.  The  relation  is  that  of  dominant  and  tonic. 

In  the  three  illustrations  under  notice  the  new  tonics  are  intro- 
duced without  the  aid  of  an  intermediate  transition  chord.  By 
including  the  signature  of  the  new  key  in  each  instance  the  author 
intended  to  imply  that  in  the  second  illustration,  for  instance,  we 


GOODRICH  S    ANALYTICAL    HARMONY. 


349 


may  consider  the  key  as  that  of  B-flat,  and  proceed  accordingly.  So 
with,  the  other  illustrations  (a)  and  (c).  In  this  sense  they  may  be 
classed  with  abrupt  transitions. 

A  more  musical  illustration  of  the  transition  to  a  major  3d  below 
is  here  quoted : 


Ex.  836. 


C.  Chaminade. 


After  the  transition  in  the  fourth  measure  the  key  of  E  is  established. 
During  the  last  two  measures  the  original  tonality  is  restored  and  the 
initial  period  in  A-flat  follows. 

Another  parallel  relationship  exists  between  any  major  tonic  and 
the  key  situated  a  large  3d  above.* 

Beethoven  established  this  in  actual  practice,  and  with  the  most 
artistic  results.  Witness  the  following:  Sonata  in  G,  Op.  31,  No.  i, 
second  theme  in  B  ;  Sonata  in  C,  Op.  53,  second  theme  in  E;  Op.  2, 
No.  3,  the  adagio  is  located  a  major  3d  above  the  main  key,  and  the 
first  and  fifth  piano  concertos  have  the  same  peculiarity.  (The  pas- 
sage to  the  major  3d  below  in  a  minor  key  was  used  by  Scarlatti  and 
Mozart  before  Beethoven  was  born.  But  this  is  a  diatonic  connection 
and  produces  a  different  effect.) 

These  two  parallel  keys  correspond  to  one  another ;  for  in  passing 
to  the  major  3d  above  we  would,  in  returning,  find  the  major  3d  below 
to  be  the  original  key-tone.  This  is  illustrated  here : 


*  Hauptmann  and  Riemann  have  remarked  upon  tHis  relation,  but  without  deducing  any 
principle  of  particular  service. 


350 


GOODRICH %S    ANALYTICAL    HARMONY. 

fe5 


Lassen. 


^=?=  =F=*=*= 

^^^£=*==*=3 


Ex.  837. 


The  quotation  begins  at  the  end  of  a  period  in  A-flat,  but  when  the 
transformation  is  made  to  C-major  at  (a)  the  key  is  treated  as  such. 
At  (c)  the  transition  is  to  a  major  3d  below,  and  here  the  original 
tonality  is  restored.  So  much  for  the  theoretical  process.  From 
an  esthetical  standpoint  the  design  is  suggestive.  During  the  first 
change  of  key  the  words  are  "  I  see  thy  form  in  mist  and  cloud  that 
over  my  pathway  rise."  Then  for  the  remainder  in  the  original  key, 
"  Thou  still  art  near,  in  darkest  night  the  stars  speak  of  thy  bright 
eyes."  The  changes  in  tonal  foundation  from  higher  to  lower  planes 
are  very  adroitly  managed,  and  the  chief  effects  are  owing  to  these 
causes. 

And  thus  it  is  that  while  merely  the  mechanical  means  at  a  com- 
poser's command  are  usually  shown,  there  yet  remains  the  more 
interesting  consideration  of  motive.  With  this  great  problem  the 
young  composer  must  sooner  or  later  concern  himself,  and  the  author 
has,  therefore,  frequently  directed  attention  to  these  unwritten  laws. 

The  direct  passage  to  the  key  a  major  3d  above  would  appear 
like  this : 


GOODRICH'S  ANALYTICAL  HARMONY. 


No  further  preparation  is  necessary.     Another  parallel  key,  some- 
what remote,  may  be  connected  a  minor  3d  below 


Ex.  839. 


Beethoven,  in  his  Op.  7,  made  this  change  from  the  allegro  to  the 
largo. 

The  following  harmonic  changes  are  effected  with  the  5th  as  a 
connecting  link : 


Ex.  840. 


The  transformation  at  (a)  bears  the  same  relationship  above  the 
original  key-tone  that  D  to  B  does  below;  i.  e.,  a  minor  3d.  One  is 
connected  through  the  3d,  the  other  through  the  5th.  These  may  be 
made  to  alternate  like  the  example  from  Lassen.  At  (b)  the  change 
is  to  dominant  minor;  an  unusual  mutation,  but  not  an  unnatural 
one.  Moszkowski  includes  it  among  the  group  of  related  keys  in 
Spanish  Dances,  Op.  12. 

The  following  has  been  given  as  a  "3d  relation": 


Ex.  841. 


But  this  progression  is  not  good,  unless  the  parallel  major  of  Z># 
•minor  intervenes. 

The  most  important  of  the  secondary,  or  transitional,  relations 
have  been  indicated.  These  are  all  that  can  be  directly  connected 
through  the  tones  of  an  original  major  chord. 

The  keys  located  a  minor  or  a  major  2d  above  or  below  any  tonic 
have  no  natural  connection,  and  in  actual  composition  they  have 
generally  been  ignored.  A  transition  to  any  of  these  four  keys  may 
be  effected,  but  we  can  not  pass  to  them  directly  with  any  hope  of 
connection  or  relationship.  Neither  is  the  key  located  an  augmented 


352 


GOODRICH'S  ANALYTICAL  HARMONY. 


4th  above  to  be  considered  as  related,  though  some  writers  so  regard 
it.  Even  if  the  mode  be  made  minor  it  seems  strangely  incongruous. 
Some  preparation  is  necessary ;  in  which  case  we  may  pass  to  A-flat- 
major  as  naturally  as  to  G%  minor  : 


a      ix-  r       .^  r       L    .         :   *.    * 
iSp^dz^z^b^nl^r-g-fr— i 

J^F-*—=    H^-?^=fp=* 


Ex.  842. 


This  might  serve  a  purpose ;  but  if  we  present  the  two  tonics  con- 
secutively there  is  not  the  slightest  connection  : 


Ex.  843. 


Such  progressions  are  esthetically  impossible,  unless  G%  minor  be 
associated  with  its  parallel  major,  B. 

A  summary  of  the  keys  related  to  a  given  major  tonic,  both  nat- 
ural and  transitional,  is  here  represented : 

NATURAL  CONNECTIONS   (DIATONIC.) 

zP^ — 


Ex.  844.  b|t=Zi 


1.  2.  3.  4. 

TRANSITIONAL  CONNECTIONS    (CHROMATIC.) 


10. 


11. 


These  should  be  written  in  A,  B-flat  and  E-flat,  with  a  separate  staff 
for  the  base,  which  is  to  be  fundamental.     *     *    * 

The  main  conclusions  to  be  drawn  from  the  preceding  apply  more 
particularly  to  the  different  movements  of  a  connected  work,  and  to 


GOODRICH'S  ANALYTICAL  HARMONY. 


353 


the  important  divisions  of  each  movement.  Chord  relations  and  key- 
relations  are,  therefore,  intimately  associated  with  Form,  especially 
that  part  which  relates  to  the  main  outlines  and  their  tonalities.  (This 
is  set  forth  in  Chapters  L,XYIII  and  L,XX.) 


ENHARMONIC  TRANSITION. 


In  the  FO-called  "  Emperor  Concerto"  Beethoven  introduced  the 
slow  movement  in  the  parallel  key  a  major  3d  below.  As  this  would 
require  for  its  signature  seven  flats,  the  composer  substituted  five 
sharps,  with  the  same  result,  practically : 


Ex.  845. 


v   7  I  *^T^rr  u 


The  enharmonic  representation  at  (b)  is  simpler,  and  therefore  pref- 
erable. 

The  greatest  enharmonic  possibilities  are  contained  in  the  dimin- 
ished yth  chord.  These  have  been  explained. 

When  the  appearance  of  a  diminished  yth  chord  is  altered  for  the 
purpose  of  establishing  a  certain  tonality  the  notation  becomes  a 
matter  of  necessity,  not  of  convenience  and  simplicity.  But  in  the 
next  example"  we  may  use  nine  flats,  or  three  sharps : 


Ex.  846. 


If  the  harmony  of  the  second  measure  appeared  transiently,  and 
returned  immediately  to  G-flat-major,  this  notation  would  be  proper. 
But  should  an  entire  period  follow  in  this  new  scale  it  would  be 
better  to  use  the  enharmonic  equivalent : 


Ex.  847. 


GOODRICH'S  ANALYTICAL  HARMONY. 


Compare  (a)  with  (b).     Also,  see  No.  i  of  Grieg's  Waltzer  Caprices, 

Op.  37- 

Another  means  of  arriving  at  any  key,  however  remote,  is  fur- 
nished by  the  chromatic  scale.  A  number  of  these  half  steps  in 
succession  have  a  tendency  to  disturb  the  tonality,  or  at  least  to 
leave  us  in  doubt  as  to  the  actual  key-tone.  (In  a  previous  chapter 
this  was  more  plainly  set  forth.) 

The  chromatic  passage  may  lead  to  the  tonic,  or  to  any  part  of 
the  new  scale.  An  instance  is  found  in  the  rondo  to  Chopin's  E-mi- 
nor  Concerto,  in  which  the  unusual  change  from  E  to  E-flat  is  accom- 
plished in  the  manner  described  : 


Ex.  848. 


At  (a)  the  tonality  of  E  is  not  .disturbed ;  but  when  the  diatonic  is 
succeeded  by  the  chromatic  scale  at  (b)  we  are  prepared  for  any  new 
key  the  fancy  of  the  composer  may  suggest.  The  rhythm,  and  the 
natural  tendency  of  the  cadenza  toward  b-flat,  aid  us  somewhat  in 
anticipating  the  actual  result  at  (c).  After  a  short  period  of  this 
principal  rondo  theme  in  E-flat  the  composer  lowers"  the  major  3d 
and  returns  almost  imperceptibly  to  the  original  key-tone.  The 
effects  are  as  charming  as  the  means  are  simple. 

REMOTE  TRANSITIONS. 

These  take  place  when  no  preparatory  chords  are  used  to  intro- 
duce the  new  key.  Such  an  instance  is  the  following  f;om  Beet- 
hoven's Op.  27,  No.  2  : 


Ex.  849. 


GOODRICH  S    ANALYTICAL    HARMONY. 


353 


From  G%  minor  to  A-major  is  not  a  very  remote  modulation,  but 
the  unexpected  manner  in  which  it  appears  strikes  one  with  almost 
tragic  force. 

The  quotation  previously  made  from  Grieg's  Op.  35  may  be  in- 
cluded under,  this  heading.  The  scheme  in  a  condensed  form  appears 
like  this : 


Ex.  850. 


These  represent  four  tonalities.     They  are  all  parallel  keys,  related 
through  the  minor  3d. 

The  King's  first  solo  in  Lohengrin,  and  the  ensemble  that  fol- 
lows, present  some  excellent  illustrations  of  this  subject.  A  few 
are  quoted : 

Ex.  851. 


By  means  of  the  4th  resolution  of  a  dominant  yth  the  music  passes 
to  D-flat,  thence  to  G-flat,  and  back  to  the  key-tone  through  the 
dominant. 

The  next  are  similar : 


Ex.  852. 


Wagner. 


Observe  the  enharmonic  change  in  the  last  measure  but  one. 

For  the  remainder,  students  who  are  ambitious  must  examine  the 
works  of  high-class  modern  composers,  where  abundant  illustrations 
may  be  discovered. 


35* 


GOODRICH'S  ANALYTICAL  HARMONY. 


Chapter  LXVII. 


ALTERED  CHORDS.     DOUBLE  AND  TRIPLE 
DISSONANCES. 


ALTERED  CHORDS. 


A  PRELIMINARY  knowledge  of  these  subjects  has  been  acquired 
-£-«L  in  former  lessons.  But  there  are  two  different  results  to  be  ob- 
tained from  altering  chords,  and  these  must  be  considered  separately. 

The  various  augmented  6th  chords  may  be  mentioned  here  as 
transition  harmonies  ;  they  are  all  altered  chords,  and  not  treated  as 
fundamental  harmonies.  But  the  majority  of  altered  chords  are  the 
result  of  a  passing  note.  Both  objects  will  appear  in  the  following 
examples. 

A  yth  chord  V  is  selected.  If  the  root  be  sharpened  the  result 
will  be  a  discord  equivalent  to  III  ;  but  as  it  leads  in  a  different  direc- 
tion it  may  be  included  here  : 


Ex.  853. 


This  is  a  passing  harmony  with  a  slight  transitional  tendency  toward 
the  dominant.  There  would  be  no  object  in  augmenting  the  major 
3d ;  but  it  may  be  lowered : 


Ex.  854. 


This  can  be  re-arranged  with  the  same  general  result. 


GOODRICH'S  ANALYTICAL  HARMONY. 


357 


The  5th  may  also  be  lowered  in  connection  with  the  3d : 


Ex.  855. 


These  are  mere  passing  notes  and  do  not  affect  the  tonality. 

Now  raise  the  5th  and  resolve  the  discord  in  any  of  the  following 
Avavs : 


Ex.  856. 


1    u  '    i  -J— I— r — I     n!    .    I     ,,1    . — I     ..   I    i  -J 1 — i 


m 


£ 


1 


Here  again  the  chromatic  alteration  is  a  passing  note.  The  base 
assumes  the  character  of  a  pedal-note  in  these  instances. 

The  corresponding  discord  upon  the  fourth  of  the  scale  is  sus- 
ceptible to  the  same  alterations.  Both  are  harsh  and  require  some 
preparation. 

The  discord  on  the  second  of  the  scale  is  next  in  order.  By 
raising  the  root  there  will  appear  a  transition  chord  with  which  we 
are  familiar  in  an  inverted  form.  But  it  is  sometimes  used  in  its 
original  position : 


Ex.  857. 


(See  overture  to  Oberon  ;  also  the  Allegro  Vivacissimo  in  Mendels- 
sohn's Scotch  Symphony.)  The  effect  is  transitional,  like  that  of 
the  augmented  6th  chord,  No.  i. 

In  the  next  example  the  root  is  lowered  as  a  passing  note : 


etc. 


Ex.  858. 


This  presents  an  unusual  combination :   major  3d,  augmented  5th, 
and  major  yth. 


35* 


GOODRICH'S  ANALYTICAL  HARMONY. 


By  lowering  the  5th  we  produce  a  discord  identical  in  appearance 
to  No.  Ill,  founded  on  the  leading  note  to  A-flat,  but  very  different 
in  its  natural  progression  : 


Ex.  859. 


At  (a)  the  d-flat  is  a  mere  passing  note  between  6  and  5  of  the  major 
scale ;  at  (b)  the  tonality  of  A-flat  is  supposed  to  have  been  estab- 
lished. If  we  desired  to  pass  into  F-minor  the  altered  interval,  as 
minor  6th,  would  serve  a  different  purpose  in  preparing  the  ear  for 
a  change  of  mode : 

' (2. 


Ex.  860. 


etc. 


At  (a)  the  d-flat  does  not  appear  as  a  passing  note.     At  (b)  the  key 
of  F-minor  is  sufficiently  established. 

The  yth  chord  III  presents  some  features  for  alteration.     Lower 
the  3d  first : 


Ex.  861. 


This  is  a  passing  harmony. 

The  3d  and  yth  may  be  lowered,  and  treated  as  in  the  following 
example  : 

ii 


Ex.  862. 


?  \y(?      &  °  r  &  \yi      *y 

1  -  L,  -  L_)  -  1          '  | 


in 


GOODRICH 'S    ANALYTICAL    HARMONY. 


359 


The  succession  of  normal  fourths  is  not  very  euphonious,  but  with 
the  3d  or  5th  uppermost  the  examples  might  be  utilized. 

The  dominant  jth  chord  is  susceptible  to  considerable  alteration. 
The  5th  is  frequently  augmented,  and  in  this  form  it  has  the  strength 
and  decision  that  characterizes  all  augmented  6th  combinations.  (See 
Exs.  510  and  511.) 

The  5th  might  be  lowered  as  a  passing  note,  thus : 


Ex.  863. 


But  this  does  not  seem  to  be  of  much  utility. 

The  flattened  3d  offers  greater  advantages,  especially  if  the  object 
is  a  transitional  one : 


Ex.  864. 


The  e-flat  leads  naturally  to  d,  and  is  characteristic  of  G -minor. 

The  jth  is  sometimes  raised  as  a  passing  note  without  destroying 
the  effect  of  the  essential  harmony : 


Ex.  865. 


Sp=«=5=i=^i 
iEtFjtz=iB=i: 


The  bb  is  a  melodic  passing  note  and  might  be  included  with  b-ftat 
as  harmonic  note  below. 

In  the  following  melodic  sequence  all  the  altered  chords  arc  pass- 
ing harmonies : 


360 


GOODRICH'S  ANALYTICAL  HARMONY. 
:fc=i^=zi 


Ex.  866. 


The  augmented  triad  at  (d)  corresponds,  in  the  harmonic  sequence, 
to  the  passing  diminished  chords  at  (a),  (b)  and  (c).  None  of  the 
chromatics  are  to  be  considered  as  transitional. 

The  following  example  illustrates  more  plainly  the  difference 
between  a  passing  chord  and  a  transition  chord: 


Fink. 


Mendelssohn. 


Ex.  867. 


_Mbi 


f^t .MC ,     -y-yr^ 

U±=^=:^r    g    1,1?    L.'^= 
^    I? 


At  (a)  the/"#  is  a  passing  note  and  does  not  affect  the  key  of  B-flat. 
At  (b)  the  /#  forms  part  of  the  essential  harmony  on  D,  and  this 
modulates  decidedly  to  G-minor. 

DOUBLE  DISSONANCES. 

With  exception  of  the  principal  Qth  chords,  nearly  all  double  dis- 
sonances are  a  product  of  suspension,  or  of  the  harmonic  appoggia- 
tura.  The  double  dissonance,  as  its  name  implies,  is  a  combination 
embracing  two  dissonating  intervals,  generally  requiring  separate 
resolution. 

Suppose  this  to  be  a  model : 


Ex.  868. 


Suppose,  farthermore,  that  the  root  of  the  first  chord  be  suspended 
after  the  discord  is  introduced  : 


GOODRICH  S    ANALYTICAL    HARMONY. 


Ex.  869. 


At  (b)  we  hear  a  double  dissonance  of  suspension — d  and  c,  and  £- 
and/.  Observe  that  each  dissonance  is  resolved  to  a  consonance,  as 
though  they  had  appeared  in  this  form  : 


Ex.  870. 


•&      & H^ 

\\      F 


The  upper  discord  at  (b)  resolves  to  a  consonant  interval  at  (c) ;  the 
lower  discord  at  (c)  is  resolved  at  (d).  The  preparation  may  be  seen 
at  (a),  Ex.  869. 

The  3d  of  the  concord  may  be  suspended  with  the  same  general 
results : 

— ^ — <.  " 


Ex.  871. 


This  is  slightly  more  dissonant  than  Ex.  869,  but  should  be  analyzed 
in  the  same  manner. 

While  it  is  true  that  a  2d  or  a  yth  may  resolve  up  as  at  (a)  in  the 
next  example  (because  the  e  is  necessary  to  complete  the  C  chord), 
the  usual  tendency  of  the  suspension  is  to  descend,  as  at  (b) : 


&  — 

I  —  <s> 

<&    f  — 

& 

- 

—ri  ^ 

^r 

~ 

K&              1 

sB 

^ 

£ 

TT 

i 

\ 

\ 

1 

> 

^2\  • 

^ 

•   1  • 

S 

*J 

Ex.  872. 


In  the  latter  instance  c  descends,  because  b  is  wanting  in  the  domi- 
nant jth  chord.     This  is  also  its  natural  resolution  here. 


362 


GOODRICH'S  ANALYTICAL  HARMONY. 


In  the  next  example  e  descends  to  d,  because  d  is  the  tone  antici- 
pated, having  been  delayed  by  the  prolongation  of  e : 


Ex.  873- 


12) 

And  since  the  upper  parts  contain  the  root,  3d  and  7th  of  the  essen- 
tial discord,  this  resolution  to  the  5th  becomes  all  the  more  imper- 
ative. 

These  double  dissonances  are  very  useful,  and  may  be  introduced 
into  the  strictest  styles  of  composition,  for  they  freely  admit  of  re-ar- 
rangement and  inversion. 

Principal  9th  chords  are  an  exception  to  this  theory.  They  fre- 
quently resolve  direct  to  a  concord,  thus,  from  Schumann : 


Ex.  874. 


The  root  and  7th,  and  the  3d  and  Qth,  form  dissonating  intervals ;  or 
they  may  be  considered  in  this  way  : 


Ex.  875. 


In  Ex.  874  the  dissonances  are  so  far  removed  from  the  fundamental 
that  they  do  not  sound  harsh ;  and  the  fact  is  to  be  considered  that 
this  is  a  principal  discord. 

In  the  following  extract  an  altered  Qth  chord  is  treated  as  a  prin- 
cipal discord : 

Mac  Powell.     Op. 


Ex.  876. 


GOODRICH'S  ANALYTICAL  HARMONY. 


363 


This  contains  a  major  3d,  augmented  5th,  minor  yth  and  minor  gth, 
an  unusual  combination.  All  the  "upper  parts  ascend  and  descend 
chromatically  while  the  base  moves  fundamentally. 

The  secondary  gth  chords,  though  a  product  of  preparation  rather 
than  of  suspension,  are  to  be  treated  as  double  dissonances  and  re- 
solved to  some  single  discord : 


Ex.  877. 


One  dissonance  disappears  at  (b)  9-8;    the  other  disappears  at  (c). 
The  following  secondary  discords  are  conducted  according  to  the 
same  principles : 


Ex.  878. 


9  resolves  to  10,  7  to  6,  and  so  on.     (See  a  to  b,  b  to  c,  c  to  d,  and 
d  to  e.) 

TRIPLE  DISSONANCES. 

Some  combinations  require  more  than  two  resolutions,  though 
these  are  of  rare  occurrence.     Such  an  instance  is  this  : 


Ex.  879. 


There  are  four  discords  here,  but  the  f-sharp  in  the  first  measure  may 
be  considered  a  passing  note.  The  combination  at  (a)  is  theoretically 
a  triple  dissonance,  since  everything  points  toward  the  dominant  jth 
harmony,  and  the  <r,  as  well  as  the  <?,  is  dissonant  to  that  chord.  At 


364 


GOODRICH'S  ANALYTICAL  HARMONY. 


(b)  we  hear  a  double  dissonance.  This  is  resolved  paitiaUy  at  (c) 
and  completely  at  (d).  Perhaps  the  design  would  be  more  readily 
grasped  if  arranged  in  this  manner : 


fir—  r- 

—  f»  — 

)     & 

~~2? 

_-«.        1 

r 

r  i   i 

»            b        c             d 

^i  — 

a 

Ex.  88o<z 


The  first  discord  might  have  been  resolved  directly  to  the  essential 
7th,  but  the  partial  resolution  at  (b)  is  more  progressive  and  gener- 
ally more  musical. 

In  relation  to  the  derivation  or  preparation  of  these  extremely 
dissonant  combinations,  the  harshness  is  more  consistent  when  they 
are  approached  by  means  of  small  steps.*  Compare  this 


Ex.  88o£. 


H 

£2Hll 


with  Ex.  880  (a),  and  it  will  be  found  less  agreeable. 
Transpose,  but  do  not  invert,  Ex.  879.     *    *     * 
A  triple  dissonance  containing  the  interval  of  an  nth  is  fre- 
quently employed.     This  has  been  called  the  "chord  of  the  nth"; 
but  it  is  a  product  of  suspension,  and  not  to  be  considered  as  a  fun- 
damental harmony.     The  usual  method  is  to  retain  the  5th  and  7th, 
and  resolve  the  gth  and  nth  down  a  second  each.     An  illustration 
is  quoted  from  Ph.  Scharwenka's  Polish  Dances : 

Ex.88i.  Op.  38,  No  2. 


*  Dissonance  may  be  produced:  i,  by  adding  to  a  concord  ;  2,  by  suspension  ;  3,  by  chro 
malic  alteration  of  a  concord  ;  4,  by  anticipation  ;  5,  by  including  an  appoggiatura. 


GOODRICH'S  ANALYTICAL  HARMONY. 


365 


The  nth  chord,  so-called,  is  prepared  by  the  secondary  yth  chord  at 
(a).  This  is  merely  repeated  upon  the  dominant,  and  after  the  figure 
above  has  completed  its  course  the  triple  dissonance  resolves  directly 
to  the  essential  discord.  The  simplest  explanation  of  the  combina- 
tion at  (b)  is  to  consider  the  base  a  pedal-note  upon  which  the  sec- 
ondary harmony  is  supported  until  it  is  merged  into  the  dominant 
7th  chord  at  (c). 

An  nth  chord  with  major  9th  is  here  quoted  from  Mascagni : 

"Cazialleria  Rusticana." 


Ex.  882. 


This  results  from  suspension.  The  dominant  is  added  to  the  base 
in  the  second  measure.  Observe  the  interval  of  a  loth  between  the 
answering  voices ;  also  the  alternate  interval  of  a  5th  between  the 
first  and  second,  third  and  fourth,  and  fourth  and  fifth  voice-parts. 
In  the  stretto  to  one  of  his  waltz-caprices  Grieg  employs  this  com- 
bination in  a  different  manner : 


fl            r7'  — 

:-L_             JL. 

N      i 

rV-  :  ^  —  r  K-i 

—  «  — 

^T~,*~*^\ 

y= 

^        PP      =-  ^ 

X^« 

&> 

JO. 

tt=P-r  ~ 
-2.         ' 

H2  
I 

c|;   4               *— 

& 

& 

-*  

fES  —      *    ' 

*> 

& 

1= 

Op.  37,  No.  2. 


Ex.  883. 


This  is  prepared  by  and  resolved  to  the  tonic  harmony.  With  excep- 
tion of  the  base  B,  the  upper  harmony  has  somewhat  the  effect  of  an 
after-cadence.  This  is  counteracted  and  made  more  decided  by  the 
dominant  below.  The  dual  character  of  this  combination  produces 
a  peculiar  effect  here. 


GOODRICH'S  ANALYTICAL  HARMONY. 


THE    DIAPAPON. 

There  is  a  still  more  dissonant  and  extended  harmonic  mass,  con- 
sisting of  every  tone  in  the  scale.  We  may  term  this  a  Diapason, 
though  it  is  known  theoretically  as  the  "  chord  of  the  i3th."  It 
results  from  suspension,  and  rests  upon  the  tonic.  In  major  this 
combination  would  be  prepared  and  resolved  in  some  such  manner 
as  this : 

3$JF^ 
Ex.  884. 


At  (b)  the  major  Qth  is  added  to  the  dominant  7th  harmony  on  the 
tonic  pedal.  At  (c)  the  base  parts  are  resolved  first  to  the  full  tonic 
chord,  while  the  upper  harmony  is  suspended  and  resolved  afterward 
at  (d).  The  diapason  at  (c)  includes  every  note  in  the  D-major  scale. 
This  is  an  elaboration  of  the  dominant  yth  chord  suspended  over  that 
of  the  tonic.  The  complete  representation,  as  here  given,  is  of  rare 
occurrence  in  the  major  scale.  In  the  next  example  this  mass  ap- 
pears with  the  3d  of  the  tonic  omitted : 

Gavotte  from  "Otho  Visconti.'1'     Glettson. 

?&  i  ^^jjiliy* 

*    y~p— *_*    #          n!          I          ^r~        I 


Ex.  885. 


— 


The  Diapason  at  (c)  resembles  an  eleventh  chord  on  the  dominant ; 
but  as  the  pedal-note  is  tonic  we  recognize  the  upper  combinations 
as  dissonances  resolving  to  a  concord  of  which  the  root  and  5th  are 
foundation  notes. 

A  remarkable  instance  in  minor  is  here  quoted  from  Beetiiovec's 
last  symphony : 


SCKTDIHCH'S   ANALYTICAL    HARMONY. 


Ex.  886. 


This  serves  to  introduce  the  baritone  solo.  The  only  preparation 
consists  of  the  dominant  in  the  kettle-drum  part,  and  the  fact  that 
the  key  was  previously  decided  as  that  of  A-major. 

The  combination  is  enumerated  from  the  tonic,  D,  thus :  I,  3,  5, 
7,  9,  n,  13.  It  is  the  diminished  yth  harmony  suspended  over  that 
of  the  tonic.  One  peculiar  feature  is  the  inverted  base.  But  this 
almost  immediately  moves  to  other  notes  of  the  chord,  whereas  the 
root  above  remains  stationary.  That  this  is  treated  as  a  suspension 
(though  there  is  very  little  preparation)  appears  more  plainly  in  the 
second  measure,  where  the  dissonances  are  resolved  to  the  tonic  har- 
mony. 

This  harmonic  mass  is  really  a  quintuple  dissonance. 


GOODRICH  S   ANALYTICAL    HAKMuJN  ,. 


PART  XVII 


Chapter  LXVIII. 


MUSICAL  FORM  AND  CONSTRUCTION. 

DIAGRAMS  OF  ELEMENTARY  MODELS. 


•*~T"VHESE  subjects,  with  their  various  divisions  and 

-1-  would  require  a  volume  for  their  adequate  explanation.  This 
system,  however,  would  not  be  complete  without  a  synopsis  of  form 
and  analysis  such  as  the  author  has  found  to  be  of  the  greatest  beiie- 
lit  to  the  average  student. 

Supposing  the  subject-matter  of  the  preceding  pages  to  have  been 
mastered,  the  next  question  is,  How  can  this  material  be  utilized? 
Form  and  Analysis  furnish  the  answer  :  The  first  embraces  outline  ; 
the  second  includes  all  the  details  of  composition.  Form  is  the  shape 
and  structure  of  anything,  as  distinguished  from  the  material  of  which 
it  is  composed.  All  the  previously  acquired  information  is  to  be  con- 
sidered as  the  material  from  which  music  is  constructed. 

MOTIVE  AND  SEMI-PHRASE. 

The  musical  motive  is  to  be  considered  as  a  subject,  or  text,  and 
tne  composition  should  be  an  outgrowth  of  this.  Short  motives 
include  but  one  measure,  and  as  these  are  the  smallest  analytical 
divisions  the  author  terms  them  Semi-  phrases.  Such  are  the  fol- 
lowing : 

Ex.  887. 

Haendel.  Mozart.  Schubert.  Chopin. 


GOODRICH'S  ANALYTICAL  HARMONY.  369 

The  small  notes  in  the  first  extract  are  included  merely  to  complete 
the  measure.  The  motive  consists  ol  the  three  repeated  notes,  .sup- 
posed to  represent  fire!  fire!  fire! 

PHRASE. 

This  contains  two  semi-phrases,  as  may  be  seen  in  the  continua- 
tion of  the  requiem  motive  : 

^          Adagio. 

Ex.  888.  " 


The  phrase  has  three  features  to  be  analyzed:  proportion,  rhythm^ 
and  melody. 

The  student  must  be  familiar  with  the  analytical  divisions  (phrase, 
section,  period,*  etc.)  and  the  different  methods  of  constructing  these. 
Considerable  practice  of  this  kind  is  necessary,  for  melodic  invention 
and  thematic  development  are  among  the  first  artistic  requisites  of  a 
composer. 

To  this  end  the  following  course  is  suggested  :  Select  the  first 
phrase  of  some  natural  melody  and  endeavor  to  supply  the  remainder 
of  the  period  without  consulting  the  original  theme.  (A  volume  of 
popular  songs,  or  the  vocal  etudes  of  Concone,  would  answer  this 
purpose.)  Then  copy  the  first  phrase  of  the  second  period  and  work 
this  out  in  the  same  synthetical  manner. 

NOTE. — The  various  methods  of  building  up  sections  and  periods  are  fully 
set  forth  in  the  author's  Musical  Analysis.  This  work  is  acknowledged  to  be 
complete  and  explicit  in  these  respects,  and  as  it  was  intended  as  a  compendium 
to  the  harmony  treatise,  no  apology  seems  necessary  for  recommending  the 
former,  especially  since  its  peculiar  field  is  not  occupied  by  any  other  text-book. 

The  outlines  of  some  of  the  smaller  forms  are  here  represented 
by  means  of  diagrams : 


-The  definitions  of  musical  phrase,  section  and  period  are  here  retained  on  account  of 
their  general  acceptance,  and  because  they  seem  to  the  author  sufficiently  appropriate 


370  GOODRICH'S  ANALYTICAL  HARMONY. 

Diagram  A. 


1st  Period. 


Ill  3d  Period.  I 


Coda. 


This  would  begin  and  end  with  the  principal 


key,  only  a  transient  modulation  being  necessary. 

Diagram  B. 

1st  Period  in  G.  I  :  2d  Period  Ditto. 


I  Trio  in  some  related  key.  I 1      D.  C.  al  /TV. 


The  tonality  here  would  not  change  materially  during  the  first  two 
periods,  especially  since  the  2d  period,  as  final  ending,  must  terminate 
upon  the  tonic  of  the  principal  key.  In  the  trio  a  different  tonality 
prevails  as  a  contrast  to  the  first  two  periods.  Some  melodic  and 
rhythmic  variety  are  also  desirable  here.  The  da  capo  is  necessary 
in  order  that  the  tonality  of  G  may  leave  its  final  impress.  An  im- 
portant element  of  form  is  proportion,  and  this  requires  that  the  repe- 
tition signs  be  disregarded  after  the  D.  C.  The  last  is  a  dance  form. 

Diagram  C. 

PART  I  IN  D. 

T\ 

1st  Period.  2d  Period. 


PART  II  IN  G  OR  Bi>. 


3d  Period.  lilt  Period 


D.  C.  al  ^. 
Part  II  is  generally  misnamed  "  trio."     See  trio  in  Diagram  B. 

Many  of  the  common  dance  species  are  built  on  the  plan  of  Dia- 
gram C.  Temporary  modulations  may  be  freely  indulged.  Among 
the  related  keys  preference  should  be  given  the  dominant,  and  the 


GOODRICH'S  ANALYTICAL  HARMONY.  371 

relative  minors  of  the  dominant  and  the  tonic.  The  subdominant  is 
usually  reserved  for  the  trio  or  coda  on  account  of  its  retrogressive 
tendency.  The  main  conditions  to  be  carried  out  are  that  the  begin- 
ning and  ending  shall  be  on  the  principal  key,  and  that  the  scale 
of  this  key  shall  be  heard  more  frequently  than  that  of  any  other. 
Hence  the  D.  C.  is  necessary  whenever  the  trio  or  part  II  are  written 
in  a  different  scale.  (Observe  the  difference  between  the  diagrams 
marked  D.  C.  and  those  which  do  not  return  to  the  beginning.) 

Diagram  D. 


1st  Period,  tonic. 


2d  Period,  related  key.  1st  Period,  tonic. 

Diagram  E. 

PART  I. 

Imt  Period.  Ill  2d  Period. 


PART  II.    SAME  SIGNATURE. 


3d  Period.  1th  Period,  tonic. 

These  are  rococo  forms,  but  still  useful. 

Diagram  F. 


1st  Period  in  C. 


2d  Period,  related  key.  3d  Period  in  C. 

Diagram  Q. 

PART  I.    C  MINOR. 


372  GOODRICH'S  ANALYTICAL  HARMONY. 

PART  II.    C  MAJOR. 


D.  C.  a!  rrv. 

Diagram  F  requires  a  closer  affinity  between  the  three  succeeding 
periods.  The  alternation  of  minor  and  major  in  Diagram  G  is  revers- 
ible. In  either  instance  the  tonic  remains  the  same.  When  these 
diagrams  are  sufficiently  understood,  the  student  should  attempt  the 
composing  of  a  few  common  dances,  such  as  mazourka,  waltz  or 
galop.  This  is  a  very  useful  preliminary  practice,  and  even  a  galop 
can  be  made  interesting.  In  Musical  Analysis  nearly  thirty  species 
of  the  dance  form  are  described,  together  with  their  intermediate 
details.  Extended  and  united  periods,  intermezzo,  coda,  etc.,  are 
also  fully  illustrated. 

The  intermezzo  may  be  used  as  a  relief  to  a  frequently  recurring 
principal  theme,  or  as  a  means  of  connecting  two  dissimilar  parts 
written  in  different  scales.  See  No.  2  of  Mendelssohn's  "Songs 
Without  Words."  An  intermezzo  begins  at  the  2Qth  measure  and 
continues  to  the  4Oth,  where  the  main  theme  recurs.  (The  last  15 
measures  comprise  the  coda.)  Another  instance  occurs  in  the  Spin- 
ning Song,  No.  34.  The  song  ends  at  25,  and  is  resumed  on  the  last 
of  29  —  the  intervening  measures  being  devoted  to  an  intermezza 
founded  upon  the  figure  of  the  introduction.  Another  intermezzo 
occurs  from  56  to  64.  The  former  is  a  mere  diversion  ;  the  latter 
serves  to  connect  the  strain  ending  in  E  with  the  one  beginning  in 
C.  The  intermezzo  is  an  important  feature,  though  it  is  seldom 
employed  in  the  dance  form. 

In  ball-room  waltzes  and  other  disconnected  works,  composers  use 
what  are  called  eingange  (entrances)  as  transitions  from  one  number 
to  another.  When  the  keys  are  unrelated  the  eingang  serves  a  pur- 
pose, especially  if  the  modulatory  section  be  cleverly  managed  ;  but  it 
were  better  to  begin  abruptly  in  the  new  scale  than  preface  it  with 
an  unnatural  or  awkward  transition.  (See  the  eingange  in  Grieg's 
Waltzer  Capricen,  Op.  37.)  The  author  applies  this  term  to  all  tran- 
sitional passages  that  aim  at  the  establishing  of  a  particular  tonality. 
In  this  sense  it  is  frequently  in  form  of  an  episode.  After  the  com- 
mon dances  a  few  rococo  or  modern  classical  species  should  be  writ- 
ten.* The  saraband,  menuetto,  gavotte  and  musette,  tarantella,  bo- 
lero, avanera  and  czardas  are  interesting  species. 

*See  partitas  and  suites  by  Couperin,  Scarlatti,  Corelli,  Paradisi,  Rameau,  Bach,  Haendel, 
Purcell,  and  Haessler. 


GOODRICH'S  ANALYTICAL  HARMONY.  373 

Even  if  these  synthetical  effusions  prove  to  be  artistically  worth- 
less they  should  be  continued  until  the  student  has  acquired  that 
ready  command  over  the  material  and  technic  of  composition  which 
«very  composer  must  possess.  The  author  does  not  regret  the 
hundreds  of  pages  of  MSS.  written  at  an  early  period  of  his  musical 
career,  even  though  they  were  afterwards  purposely  destroyed,  and 
though  he  has  for  a  number  of  years  past  discontinued  all  efforts  at 
composition  It  teaches  that  which  no  book  and  no  professor  can 
impart. 


Chapter  LXIX. 

MUSICAL  FORM  AND  CONSTRUCTION 
CONTINUED. 

RHYTHM. 

~T~)  HYTHM  is  an  important  element  of  construction,  and  yet  it 
-^-  must  be  sufficiently  varied  to  prevent  monotony.  Two,  or  even 
three,  periods  may  be  built  up  from  a  single  rhythmic  design,  but  in 
the  second  part  a  different  arrangement  will  be  necessary.  Great 
care  must,  however,  be  exercised  in  the  choice  of  rhythmical  devices, 
for  rhythm  is  the  principal  element  of  dance  music.  It  represents 
action  and  motion,  and  nearly  all  characteristic  rhythms  are  suggest- 
ive of  mechanical  effort  or  physical  exertion.  Observe  the  rhythm 
and  melody  in  such  airs  as  He  shall  feed  His  flock,  I  know  that  my 
Redeemer  liveth,  or  the  Tuba  Mirum  from  Mozart's  Requiem  : 

Ex.  889. 

f  f  Voice.  - 

/  /•>  .       •  o  "T~     m 


Tu  -  ba   minim  spargeus  so  num. 

The  trombone  motive  sounded  in  advance  is  especially  appropriate 
to  the  text;  the  melody  is  consonant  to  the  sentiment,  and  the 
rhythm  and  movement  correspond  to  the  accent  and  metre  of  the 


574  GOODRICH'S  ANALYTICAL  HARMONY. 

words.  This  may  also  be  said  of  the  songs  from  Haendel.  These 
are  slow  movements,  but  fast  movements  might  be  cited  as  well. 
Rejoice,  rejoice,  from  "  The  Messiah";  the  second  part  to  Beethoven's 
Adelaide  ;  and  Di  quella  pira  from  "  II  Trovatore,"  are  instances  in 
which  rhythm  and  movement  are  sufficiently  animating  to  express 
the  sentiments  of  the  words,  and  yet  without  any  suggestion  of  danc- 
ing, marching,  or  mere  physical  exertion.  Notice  also  the  Slumber 
Song  by  Schumann,  Op.  124.  How  chaste  and  gentle  and  hopeful 
the  melody ;  how  artistic  the  harmonization,  and  how  suggestive  the 
agitation  of  the  trio  in  G-minor!  There  is  a  certain  rhythmic  move- 
ment that  accompanies  the  song,  but  this  suggests  the  motion  of  a 
cradle,  and  is,  therefore,  a  necessary  feature  of  the  song. 

J.  S.  Bach  was  a  great  master  of  rhythm,  and  displayed  rare  judg- 
ment in  its  application  ;  for  though  he  composed  considerable  music 
in  the  dance  form,  we  find  in  his  serious  works  very  little  of  terpsi- 
chorean  suggestion.  The  motive  of  one  of  his  clavier  fugues  is  quoted 
as  an  example : 

•*••*•  * 


Ex.  890.  EBtfefauT^J^P 


Intelligent  discrimination  must  be  exercised  in  this  matter,  for  in  a 
nocturne,  or  any  composition  of  a  contemplative  nature,  it  would 
certainly  be  incongruous  to  introduce  the  bustle  and  swing  of  a 
dance  rhythm.  Beethoven's  object  in  substituting  the  scherzo  for 
the  minuet  was  undoubtedly  the  eliminating  of  dance  elements  from 
pure  instrumental  music. 

Moszkowski,  in  his  Spanish  Dances,  Op.  12,  has  succeeded  in 
reproducing  not  alone  the  rhythm  but  the  national  characteristics 
with  remarkable  cleverness. 

The  student  must  understand  that  national  dances  in  some  of  the 
older  countries  are  so  characteristic  as  to  form  something  of  a  psycho- 
logical index  to  the  habits  and  aspirations  of  the  people.  Therefore 
we  who  have  no  national  dances  must  judge  from  a  more  remote 
standpoint.  We  must  know  their  origin  and  the  historic  associations 
connected  with  these  old  dances  in  order  to  utilize  them  in  artistic 
composition  ;  for  the  author  can  conceive  no  greater  musical  vulgar- 
ity than  the  introducing  of  a  common  dance-rhythm  into  a  serenade, 
overture,  sonata,  or  string-quartet.  The  young  composer  is  there- 
fore admonished  to  exclude  all  terpsichorean  rhythms  from  his  sen- 


GOODRICH'S  ANALYTICAL  HARMONY.  375 

ous  compositions  until  he  knows  their  significance  and  is  sure  of  their 
appropriate  application. 

Beethoven  introduced  in  the  scherzo  of  his  Pastoral  Symphony  a- 
rustic  dance,  and  with  charming  drollery  of  effect.  So  in  the  Italian 
Symphony,  by  Mendelssohn ;  the  Country  Wedding,  by  Goldmark ; 
Jm  Walde,  by  Raff;  Frithjof,  by  H.  Hofmann ;  Saint-Saens'  Danse 
Macabre ;  the  ideal  dances  of  Chopin,  and  such  works  as  Liszt's 
Hungarian  rhapsodies,  and  the  Op.  38  of  Ph.  Scharwenka. 

In  the  first  movement  to  Beethoven's  5th  symphony  the  rhythm 
of  the  fate-motive  is  imitated  almost  continually ;  even  during  the 
lyrical  second  theme  it  is  heard  from  the  bases 


Every  concert-goer  will  remember  how  persistently  the 

i     r- •> 
044 

in  the  allegretto  of  the  yth  symphony  is  maintained.  Another  famil- 
iar instance  is  the  Erl  King,  by  Schubert,  wherein  the  triplets  of  the 
accompaniment  are  continued  throughout.* 

In  regard  to  mensural  proportion,  extended  and  united  periods 
are  important  features,  especially  in  works  of  considerable  length. 
Introduction,  eingang,  anticipation,  intermezzo,  passage,  development 
and  termination  admit  considerable  irregularity  in  their  rhythmic 
proportions.  Lyric  movements  must  be  more  equal  and  symmet- 
rical in  their  periodic  construction. 

Here  also  may  be  mentioned  the  numerous  avoided  and  decep- 
tive cadences  which  have  a  tendency  to  prolong  the  periods  and  thus 
prevent  the  interest  from  prematurely  subsiding. 

Uneven  rhythmic  phrases,  irregular  periodic  construction  and 
continued  thesis  all  have  the  advantage  of  relieving  an  otherwise 
monotonous  movement  by  the  variety  of  effect  which  they  produce. 

Provided  the  student  is  endowed  writh  sufficient  intuitive  capac- 
ity, there  yet  remain  the  more  important  secrets  of  thematic  elabora- 
tion, choice  of  means,  and  significance  of  effect. 

Development  and  form  will  be  discussed  in  the  following  chapter. 


*  The  author's  principal  definition  of  rhythm  applies  to  the  value  and  arrangement  of 
notes  in  a  measure.  In  a  more  general  sense  rhythm  refers  to  mensural  proportion,  and 
includes  accent  and  movement. 


376  ^GOODRICH'S  ANALYTICAL  HARMONY. 


Chapter  LXX. 


MUSICAL  FORM  AND  CONSTRUCTION 
CONCLUDED. 

The  Sonata  Form  in  Major  and  in  Minor:     Outline, 

Tonality,  Development,  Affinity  of  Motives, 

Diagrams,  etc. 

OUTLINE. 

THE  sonata  is  a  cyclical  form  consisting  of  three  or  four  move- 
ments. The  first  of  these,  usually  an  allegro,  is  most  important, 
and  will  be  briefly  outlined.  This  is  founded  upon  a  formal  plan  as 
to  symmetrical  proportions,  tonal  arrangement  and  logical  develop- 
ment. There  are  three  main  divisions  to  the  first  allegro  (sonata 
movement)  : 

1 .  From  the  beginning  to  the  double  bar. 

2.  From  the  double  bar  to  the  end  of  the  "  development." 

3.  From  the  reprise,  or  return  of  the  principal  theme,  to  the  end 
of  the  movement. 

The  first  division  is  a  citation  of  the  leading  motives.  The  sec- 
ond division  consists  of  an  elaboration,  or  discussion,  of  the  principal 
motives.  The  third  division  is  similar  to  the  first,  excepting  in 
tonality. 

The  first  and  third  divisions  have  three  subdivisions:  "  Principal 
theme,"  "  Second  theme,"  and  "  Conclusion." 

The  main  theme  contains  from  16  to  60  measures,  depending 
upon  the  dimensions  of  the  work.  The  second  subject  is  about  the 
same  in  mensural  proportion.  The  conclusion  is  shortest  of  the 
three  subjects.  The  development  was  originally  brief,  but  it  has 
been  enlarged  since  the  advent  of  Beethoven.  In  his  Op.  2,  No.  2, 
the  development  contains  103  measures. 


GOODRICH'S  ANALYTICAL,  HARMONY.  377 

TONALITY. 

The  first  theme  is  principally  in  tonic  major.  The  second  theme 
and  conclusion  must,  according  to  the  old  formula,  be  in  the  domi- 
nant. The  dominant  modulation  has  already  been  explained  har- 
monically. The  author  has  also  pointed  out,  in  another  work,  that 
a  new  tonality  presents  a  different  view,  on  account  of  its  different 
location  above  or  below  the  original  tonic.  The  mere  difference  in 
signature  between  the  related  keys  is  not  what  produces  the  effect 
of  a  new  tonality.  It  is  to  be  found  in  the  metaphysical  relation  of 
keys  and  the  different  views  presented  by  the  changing  tonalities. 

The  development  may  begin  in  any  scale  that  naturally  suggests 
itself,  excepting  that  of  the  original  tonic.  The  free  use  of  transition 
becomes  a  necessity  here  in  order  to  present  the  chief  motives  in 
different  lights  and  colors. 

The  reprise  usually  recurs  in  the  tonic.  The  second  subject  is 
transposed  from  dominant  to  tonic,  as  is  the  conclusion,  so  that  the 
movement  may  end  in  the  original  scale,  thus  maintaining  the  su- 
premacy of  the  principal  key. 

The  location  of  the  second  theme  can  no  longer  be  prescribed. 
Any  of  the  parallel  keys  may  be  selected. 

DEVELOPMENT. 

This  has  an  important  influence  upon  musical  construction  in 
general.  The  difference  between  variation  and  development  must 
first  be  understood.  In  the  former  the  melodic  outline  and  harmonic 
structure  usually  remain  the  same.  In  the  latter  only  a  part  of  the 
theme  is  selected,  and  this  is  led  in  a  different  direction  from  that 
of  the  original,  being  sequenced,  modulated,  or  otherwise  metamor- 
phosed. 

Variation  shows  the  same  picture  in  different  phases;  develop- 
ment exhibits  only  a  part  of  the  picture,  and  then  presents  other 
views  of  a  kindred  nature.  (No  reference  is  here  made  to  those 
so-called  variations  in  which  the  theme  is  repeated  identically  amidst 
the  idle  flurry  of  arpeggio  and  scale  passages.  This  is  merely  varia- 
tion of  the  accompaniment,  not  of  the  melody.) 

The  first  section  of  a  theme  upon  which  Beethoven  wrote  three 
sets  of  variations  is  quoted : 


378 


GOODRICH'S  ANALYTICAL  HARMONY. 


Ex.  8gia. 

Andante. 


Op.  14,  No.  2. 


After  observing  the  melody  and  harmony  of  these  two  phrases  they 
should  be  compared  with  the  following  variation,  corresponding  to 
this  section  : 


Ex. 


N 

P 

-    > 


The  harmonic  substance  is  identical,  but  the  rhythm  and  style  are 
varied.  Observe  that  the  upper  notes,  indicated  by  double  stems, 
represent  the  original  theme. 

Variation  may  thus  be  used  as  a  means  of  construction  to  prevent 
a  repeated  passage  from  sounding  monotonous,  or  to  exhibit  a  recur- 
ring theme  in  different  colors.  (The  reader  should  here  refer  to 
Beethoven's  theme  and  variations  in  the  Sonata,  Op.  26 ;  the  last 
half  of  the  adagio  in  his  first  F-minor  Sonata ;  and  to  the  Sonata  in 
A,  by  Mozart,  No.  6,  Edition  Litolff.  Schumann's  Op.  46  represents 
a  still  more  artistic  unfolding  of  a  musical  germ,  and  belongs  to 
development  rather  than  to  variation.) 

Examples  of  development  are  now  presented : 


GOODRICH  S   ANALYTICAL    HARMONY.  379 

"Scotch  Symphony."     Mendelssohn. 


Allegro. 


Ex.  892. 


Among  numerous  transformations  of  this  motive,  in  the  develop- 
ment, notice  the  following : 


Ex.  893. 


The  natural  melodic  tendency  of  the  theme  is  not  followed  here,  but 
the  second  measure  is  a  contrary  inversion  of  the  first.    The  rhythm 


is  maintained  throughout. 


The  next  quotation  is  more  elaborate  : 


Ex.  894. 


HE  te^  Ii3=i=i=  Eclfe 


Observe  the  isolated  phrases  of  the  highest  part  in  connection  with 
the  original  theme.  Each  voice-part  in  Ex.  894  represents  a  devel- 
opment of  the  motive. 


380 


GOODRICH'S  ANALYTICAL  HARMONY. 


Such  designs  excite  the  keenest  interest  because  of  the  various 
voices  all  talking  about  the  same  subject  in  a  different  manner. 

Another  form  of  metamorphosis  is  represented  in  the  next  quota- 
tion : 


Ex.  895. 


7..n 


The  two  middle  parts  carry  on  a  free  canon,  with  ad  libitum  parts 
above  and  below.  For  other  instances  see  the  full  score,  or  the  four- 
hand  piano  arrangement. 

The  subject  of  development  will  conclude  with  a  few  excerpts 
from  Schubert's  Tragic  Symphony  : 


Ex.  896. 


Principal  Them*  of  the  l»t  Allejrro. 


-•-*- 


It  is  scarcely  necessary  to  remark  that  this  is  perfectly  natural  and 
melodious.     Now  observe  the  first  of  the  elaboration  : 


Ex.  897. 
•fe 


I   IIIHOIIO      Tulli. 


.    ,  *u   A       __--  --   ••   *•  '*-- 


ff 


GOODRICH'S  ANALYTICAL  HARMONY.  381 

This  illustrates  the  general  principles  still  more  plainly,  especially 
with  regard  to  sequence.  Only  the  first  of  the  motive  is  here  devel- 
oped. 

In  the  last  quotation  two  different  phases  of  the  principal  theme 
are  elaborated : 


Ex.  898. 


^*     ^    0    '                                    ^  * 
-    •*-•*-      —- — A -4— ^       •*- 
W. — , 1—.— =1 .— .  -  — w-—  i . ^  1 


At  (a)  the  rhythm  is  slightly  altered,  and  no  attempt  is  made  to 
pursue  the  natural  trend  of  the  melody.  At  (b)  a  smaller  fragment 
of  the  original  motive  is  taken  as  a  model,  and  this  is  continued  in 
sequence  beyond  the  quotation. 

In  addition  to  sequence  and  passage,  the  various  kinds  of  canonic 
imitation  play  important  parts  in  elaboration.  Augmentation,  dimi- 
nution, repetition,  rhythmic  imitation  and  transition  are  also  means 
to  this  end.  But  the  lessons  should  not  terminate  here.  The  student 
must  consult  standard  compositions  in  this  and  all  other  matters  that 
relate  to  musical  construction.  Enough  has  been  explained  to  enable 
the  observing  reader  to  examine  profitably  the  thematic  work  of  emi- 
nent composers,  and  they  are  the  greatest  teachers  of  the  secrets  of 
composition. 

In  Mozart's  last  three  symphonies  the  fourth  as  well  as  the  first 
movements  are  in  sonata  form.  The  finale  to  the  "Jupiter  Sym- 
phony" contains  the  most  ingenious  and  complicated  development. 
The  unraveling  of  these  musical  threads  will  sufficiently  tax  the 
mind  of  the  reader,  though  it  was  all  perfectly  easy  to  Mozart ! 

AFFINITY  OF  MOTIVES. 

A  feature  of  great  importance  now  demands  attention,  and  as  it 
is  a  more  or  less  latent  principle  the  young  composer  will  do  well  to 
give  to  it  his  most  serious  endeavor.  Reference  is  made  to  Unity  of 
design,  or  the  innate  affinity  and  relationship  of  the  different  move- 
ments to  the  original  motive.  A  symphony,  overture,  concerto, 
string-quartet,  sonata,  is  not  a  hotch-potch  medley,  but  a  congruous 


382  GOODKICH'S  ANALYTICAL  HARMONY. 

and  connected  work ;  a  logical  illustration  of  some  musical  impres- 
sion. The  motive  is  to  be  considered  as  a  subject  to  be  discoursed 
upon  and  illustrated  in  various  lights  and  colors.  Such  a  work  as 
Tschaikowski's  E-minor  Symphony  is  the  psychological  expression 
of  a  series  of  kindred  emotional  images. 

The  leading  motives  from  Schubert's  B~flat  Symphony  are  quoted. 
These  represent  the  four  movements : 


The  principal  theme  (a)  is  a  chord-motive,  and  the  various  ramifi- 
cations of  this  may  easily  be  traced  through  the  entire  symphony. 
Another  phase  of  the  subject  appears  at  the  sixth  measure,  which  is 
employed  as  a  counter-subject  during  the  repetition.  A  section  of 
the  2d  theme  appears  at  (b).  The  outline  of  this  is  also  a  chord- 
motive.  No  analytical  knowledge  is  required  in  tracing  the  affinity 
between  the  allegro  and  the  andante. 

The  minuet  contains  the  same  motive  in  different  measure,  and 
changed  from  major  to  minor.  In  the  trio  the  original  motive  is 
reversed.  The  same  coherency  is  observable  in  the  finale.  See  (f), 
(g),  and  (h;. 

Attention  is  now  directed  to  the  Sonata,  Op.  13,  by  Beethoven. 


GOODRICH'S  ANALYTICAL  HARMONY. 


333 


This  should  be  examined  in  detail,  for  the  unity  of  design  is  dis- 
tinctly traceable.  Observe  first  these  three  notes,  the  germ  of  the 
sonata: 


They  occur  in  the  very  first  of  the  allegro,  and  are  indicated  bv  accent 
marks : 

Ex.  901. 


In  the  second  subject  they  appear  in  this  form : 


Ex.  902. 


and  in  the  rondo  thus  : 


Ex.  903. 


^  ^~f 


Observe,  not  only  the  c,  d,  e-JJat,  but  the  e-flat,  d,  c,  descending,  whia  i 
is  the  original  motive  reversed.  These  tones  also  occur  in  the  epi- 
sode, and  in  the  second  theme  of  the  adagio,  though  the  latter  is 
founded  upon  the  second  half  of  the  original  motive  : 


Ex.  904. 


Another  melodic  figure,  of  a  subsidiary  character,  occurs  in  all  the 
movements  in  different  guises.     Two  of  these  are  presented : 


-""•im 


Adagrio. 

Even  this  motive  is  a  natural  outgrowth  of  the  principal  theme. 

The  suites  and  partitas  of  Corelli,  Couperin,  Scarlatti,  Paradi;;i, 
Bach  and  Haendel  contain  many  interesting  illustrations  of  coherent 
{.hematic  development  and  affinity  of  motives.  The  melodists  of  the 
{Stli  and  igih  centuries  frequently  lost  sight  of  congruity  and  iioia<> 


GOODRICH'S  ANALYTICAL  HARMONY. 


geneity,  though  the  best  composers  of  the  present  century  have  aimed 
at  greater  unity  and  connection.  Many  of  Grieg's  works  contain  but 
a  single  motive  worked  out  and  elaborated  in  the  most  concise  and 
masterly  manner.  Observe  the  funeral  march  upon  the  death  of 
Ase,  and  the  last  movement  in  the  first  Peer  Gynt  suite. 

SONATA  FORM   IN   MINOR. 

The  principal  differences  here  are  in  relation  to  mode  and  tonal- 
ity. The  classical  formula  is  as  follows  •  First  theme  in  tonic  minor ; 
second  theme  in  the  relative  major;  conclusion,  the  same.  This  last 
subdivision  is  sometimes  modulated  to  the  dominant  on  account  of 
the  repeat.  Development  in  various  tonalities ;  reprise,  tonic  minor  ; 
second  subject,  tonic  major,  or  tonic  minor ;  conclusion,  the  same. 
The  relative  major  of  the  subdominant  is  also  used  for  the  second 
theme  in  the  last  division. 

The  student  should  now  attempt  the  composition  of  a  Sonatina 
in  major,  and  one  in  minor.  To  facilitate  these  lessons  a  few  dia- 
grams are  given,  showing  the  forms  in  outline : 


Allegro  I.    A. 


SONATINA  IN  F-MAJOR. 


<• 

1                   Principal  theme  in  F.     Thematic. 
16  to  20  measures. 

Extended  period 
or  modulation. 

B. 


C. 


2d  Theme  (lyric)  in  C-major  or  A-minor. 
16  to  20  measures.* 


Conclusion. 
Same  tonality. 


D. 


Development  in  various  tonalities,  using  frag- 


D. 


ments  of  at  least  two  themes  from 
A,  B,  or  C. 

Modulation  or  cadenza 
leading  naturally  to  the 

• 

c  The  spaces  represent  jthe  relative  mensural  proportion  of  the  different  divisions  and 
subdivisions. 


GOODRICH'S  ANALYTICAL  HARMONY. 


335 


A. 


Reprise.     Same  as  first  subdivision  in  F. 


B. 


\ 

2d  theme 

in  F-major  or  D-minor. 

Conclusion  in  F. 

At  the  end  of  the  first  subject  in  the  reprise  the  modulation  .must 
of  course  be  altered. 

A  coda  is  sometimes  added  to  the  conclusion  as  final  ending.  Or 
the  conclusion  may  be  extended.  See  Beethoven's  Sonata  in  F-mi- 
nor,  Op.  2,  No.  i. 


Andante  If  (a). 


D-MINOR,   B>,  OR   Db-MAJOR. 


:                ist  Period.               : 

;                2d  Period.                : 

Coda. 

II  (b)  or  this  form  may  be  substituted  for  (a). 


Principal  theme  extended. 

Intermezzo, 
(irregular.) 

Main  theme 

repeated  and 

somewhat  varied. 

Coda. 

At  least  one  change  in  measure  should  be  included,  for  it  is  not 
well  to  have  all  the  movements  alike  in  this  respect. 

Rondo  III.     In  F. 


Principal  theme.     About  16  measures. 


Intermezzo,  irregular 
and  transitional. 


386  GOODRICH'S  ANALYTICAL  HARMONY. 


Principal  theme  as  at  first,  in  F. 


2d  theme  in  some    f   A  contrast  to  the 
parallel  key.         (        main  theme. 

Eingang  or  cadenza, 
leading  to  the 

Principal  theme  as  at  first. 

1       Termination,  or  coda 
in  F. 

The  difference  between  a  regular  period  with  coda  and  an  ex- 
tended period  may  be  illustrated  in  this  manner : 


Regular  period  of  16  measures. 

Coda  of 
4  measures. 

I 

•  Extended  period  of  20  measures. 


The  first  is  isolated,  the  second  is  continuous  and  uninterrupted. 

Irregular  intermezzo  is  intended  to  bignify  that  the  mensural  pro- 
portion is  uneven,  and  that  the  construction,  being  less  melodious 
than  the  preceding  period,  does  not  divide  itself  into  regular  phrases 
and  sections.  This  is  also  a  peculiar  feature  of  eingang,  anticipation, 
coda,  and  termination.  Tne  student  must  not  conclude  from  this  that 
rhythmical  balance  and  symmetrical  proportion  may  be  lightly  set 
aside.  Irregulai  periodic  construction  is  effective  only  as  a  relief  to 
the  even  rhythmical  balance  of  regular  periods ;  and  the  former  must 
be  so  conceived  that  the  irregularity  is  not  especially  noticeable.* 

It  will  be  a  pleasant  task  for  the  student  to  fill  in  these  outlines, 
though  some  difficulty  may  be  encountered  in  devising  motives  with 
sufficient  affinity  for  the  various  movements. 

No  diagrams  are  necessary  for  the  sonatina  in  minor,  because  the 
outlines  remain  very  nearly  the  same. 

*  These  features  are  illustrated  in  Musical  Analysis. 


GOODRICH  S    ANALYTICAL    HARMONY.  387 

With  regard  to  the  slow  movements,  the  first  form  (a)  may  be 
used  for  the  sonatina  in  major,  and  the  second  (b)  for  the  one  in 
minor. 

In  the  rondo  of  the  latter  a  recollection  and  stretto*  may  be  sub- 
stituted for  the  termination  indicated  in  the  diagram. 

(If  required,  motives  may  be  found  in  the  Key  to  this  work.) 

A  few  concluding  sentences  with  regard  to  the  manner  of  com- 
posing :  Creative  artists  do  not  strum  out  their  music  from  the  key- 
board of  a  piano  or  an  organ.  That  is  manufacturing,  not  composing 
music.  If  one  can  conceive  a  theme,  or  a  harmonic  progression,  it 
can  be  committed  to  paper  without  the  aid  of  an  instrument.  And 
the  very  conception  of  an  idea  presupposes  that  its  author  knows 
how  it  will  sound. f 

Certain  students  may,  however,  need  the  very  practice  wrhich  this 
strumming  process  affords,  and  it  might  be  advisable  for  them  to  test 
every  phrase,  section  and  period  through  the  agency  of  a  piano  or 
organ ;  at  least  until  they  can  judge  a  passage  independently  of  its 
actual  performance. 


-These  features  are  illustrated  in  Musical  Analysis. 

fHans  Richter  asserts  that  while  composing  The  Mastersin^ers,  Wagner  never  sounded 
the  piano  in  his  music-room. 


GOODRICH'S  ANALYTICAL  HARMONY.  389 


INDEX  OF  SUBJECTS. 


Most  of  the  references  are  to  chapters,  indicated  by  Roman  numerals,  or 
4o  examples  in  notation.  The  pages  are  not  given  excepting  where  a  subject 
is  merely  mentioned ;  then  they  are  indicated  by  cardinal  numbers. 

A  .  .Accent.     LI,  LIT,  LVI,  LVII. 

Accompaniment.     LXI. 

Affinity  of  Motives.     LXX.     Ex.  899. 

After  Cadence.     XLI,  XLII. 

Altered  Chords.    XXIX,  XLIU,  XLIV,  XLV,  XLVI,  LXVH. 

Ambiguous  Cadence.     Exs.  470,  471.     Also  Ch.  XLIX. 

Amen  Cadence— Its  Application.     Ex.  452. 

Anticipation.     LVII. 

Appoggiatura  (Harmonic).     LVI. 

Augmented  2d.     XVI,  XLIX,  LXII.     Ex.  796. 

Augmented  6th.     XLIII,  XLIV,  XLV,  XLVI. 

Augmented  Triad.     XXIX. 

Auricular  Exercises.     (See  Practical  Exercises.)     Page.  185. 
B  .  .Base,  Inverted.     XXVII,  XXIX,  XXXIV,  XXXV. 

Base,  Real.     XXVII,  XXIX,  XXXIV.     Exs.  698,  809  (b.) 

Base,  Suspended.     Exs.  587,  589,  742,  871. 

C  .  .Character  of  Certain  Harmonic  Progressions  Analyzed.     XXXV,  XXXVI, 
XLVI,  XLVII,  LXVI. 

Chord  Connections.     VII,  VIII,  XXII. 

Chord  Movements.    VII,  VIII,  IX,  X,  XI,  XIII,  XX,  XXI,  XXIII,  XXVIII, 
XXXVI,  LXVI. 

Chord  Relations.     XIII,  XVIII,  XXXVI,  LXVI. 

Chord  Representation.     XXI,  XXII,  LV. 

Chromatic  Harmonization.     XXXVII,  XXXIX.     Also  Exs.  748,  802. 

Chromatic  Scale.     LVII.     Ex.  848. 

Close  Position.    V,  VI. 

Comparison  of  Sub-dominant  and  Dominant  in  Relation  to  Tonic.    Pages 
220  and  221. 

Connecting  Links.     (See  Notes  of  Connection.)     VII,  XXXVI. 

Consecutive  Fifths.     (See  Parallel  Movements.) 

Consecutive  Octaves.     (See  Parallel  Movements.) 

Counterpoint :  Harmonic.     LVIII,  LIX,  LX. 

Cross  Relation.     LXII. 


3QO  GOODRICH'S  ANALYTICAL,  HARMONY. 

D  . .  Deceptive  Cadence.     Exs.  449,  450. 

Derived  Harmonies.     XXIX,  XLIII,  XLIV,  XLV,  L,  LXVII. 

Design.     (Motive,  Object.)     LV,  LVII,  LX,  LXVI. 

Development.     LXX. 

Diapason.     Exs.  884,  885,  886,  and  remarks  following. 

Diatonic  Progressions.'    XIII,  XIV,  XVI. 

Diatonic  and  Chromatic  Triad  Progressions.     Exs.  351,  844. 

Diminished  Seventh  Chord.     XXX,  XXXI,  XXXIII,  XXXIV,  XXXVIIL 
XXXIX,  XL. 

Direct  Cadence :  Authentic.     XLJ,  XLII. 

Directions  for  the  Base.     VIII,  XXVII,  XXXIV. 

Disconnected  Progressions.     XIII,  XXXVI,  XLYVI. 

Dispersed  Harmony.     LVIII,  LIX,  LX. 

Distinction  between  Variation  and  Development.     LXX. 

Dominant  Chord.     XVIII,  XX. 

Dominant  Seventh  Chord.     XXI,  XXII,  XXIII,  XXIV,  XXV,  XXXII. 
XXXVII,  LIV. 

Double  Pedal-notes.     LIU. 

Double  Appoggiatura.     Ex.  678. 

Double  Suspension.     LII. 

Drone  Base.     (See  Double-pedal.) 

Duophonic  Chord.     Exs.  24,  176,  177,  356. 

Duplication.     XXII,  XL.VII,  LV. 

E.. Effect  of  Simultaneous  Intervals  in  two-part  Counterpoint.     LIX.     Ex. 
717. 

Eingang.     LXVIII. 

Eleventh  Chords.     LXVII. 

Embellishment.     LVII. 

Enharmonic  Representation.     XXXVIII,  LXVI.     Also   Ex.  171   and  re- 
marks preceding ;  and  Exs.  846,  847. 

Enharmonic  Transition.     XXXVIII,  LXVI. 

Esthetic  Character  of  Rhythm.     LXIX. 
F  .  .False  Relation.    LXII. 

Figurated  Accompaniment.     LXI. 

Form.    LXVIII,  LXIX,  LXX. 

Fundamental  Progressions.     VIII  to  XXVI. 

Fundamental  Harmonies.     I  to  XXVIII. 
Q  .  .General  Base.     XLVIII. 

Ground  Base.     XLI. 
H  .  .Half-open  Position.     XXII,  XXVII. 

Harmonic  Minor  Scale.     XVI,  XXXVIII. 

Harmonic  Progressions.     XIII,  XVI,  XXXVI,  XLVII,  LIV,  LXIII,  LXV, 
LXVI. 

Harmonic  Tone.     LVI. 

Hidden  Fifths.     LXII. 

Hidden  Octaves.     LXII. 

Holding  Tone.     (See  Stationary  Tone.) 

How  to  find  a  Root.     VIII. 


GOODRICH 'S   ANALYTICAL   HARMONY.  391 

I    ..Imperfect  Triad.     XXIX,  XLII. 

Imperfect  Leading-tone.     XLII,  XLIX. 

Interval  Defined.     I,  XXIX,  XLV. 

Intervals  in  Parallel  Movement.     LX. 

Intermezzo ;  objects  of.     LXVIII. 

Inversion;     XXVII,  XXXIII,  XXXIX,  XLIII,  XLIV,  XLV,  LI,  LII,  LXIII. 

Inverted  Harmonies.     Exs.  5270  and  b. 
J   .  Justifiable  Fifths.     Exs.  783,  831. 

Justifiable  Octaves.     Exs.  352,  830. 
K  . .  Key  Impressions.     Exs.  394,  530,  653. 

Key  vs.  Mode.     pp.  169,  170. 
L  .  .Leading-tone.     XVIII,  XX,  XXXI. 

Lower  parts  :  Base,  Baritone,  Tenor.     LVIII,  LXV.     Exs.  823,  828. 
Al.  .Major  Concords.     III. 

Major  Resolutions  of  the  Diminished  7th.     Exs.  436,  445. 

Major  Resolutions  of  Dominant  7th.     XXI,  XXII. 

Melodic  Minor  Scale  Harmonized.     XLIX. 

Minor  Concords.     IV. 

Minor  Resolutions  of  Dominant  7th.     Exs.  547  to  552. 

Modulation.     XVIII,  XX,  XXIII,  XXX,  XLIII,  XLIV,  XLV. 

Modulation  and  Progression  Distinguished.     XVIII. 

Monotone.     XLVII. 

Motive :  Semi-phrase.     LXVIII.     Ex.  887. 
N  .  .Natural  Harmonics.     Ex.  25. 

Natural  Minor  Scale  Analyzed.     XLIX. 

Natural  Modulations.     XVIII,  XIX,  XX,  XXXI. 

Neapolitan  6th  (so-called).     Ex.  465. 

Ninth  Chords.     L,  LXVII. 

Normal  sth.     I,  II. 

Normal  4th.     I,  II. 

Normal  Major  Scale.     I. 

Notation  :  Theory  of,     XXXVII,  XXXVIII,  XXXIX.  XL. 

Notes  of  Connection.     VII,  XIV. 
O  .  .Omission.     XXII,  LV.     Also  Exs.  365,  366  and  654. 

Open  Position.     LVIII,  LIX,  LX. 

Organ-Point.     LIU,  LX. 

Outline.     LXX. 
P  .  .Parallel  Movements.     XII,  XXVIII,  LX,  LXII. 

Passing  Chords.     XXXV,  XL,  LIII,  LXVII. 

Passing  Modulations.     XVIII,  XX,  XXXI,  XXXVII.     Ex.  530. 

Passing  Tone.     LVI,  LVII,  LXVII 

Pedal-note.     LIII,  LX. 

Pedal-note  :  Necessity  for.     Exs.  607,  610,  748,  756. 

Period.     LXVIII. 

Phrase.     LXVIII. 

Preparation  of  Discords.     XXXV,  L,  LXVII. 
Q  .  .Quartet,  Vocal.     LVIII,  LIX,  LX. 

Quint  Succession.     (See  Parallel  Fifths.) 


392  GOODRICH'S  ANALYTICAL  HARMONY. 

R  .  .Real-Base.     XXVII,  XXIX,  XXXIV,  LV. 

Related  Keys.     XVIII,  XX,  XLVII,  LXVI. 

Remote  Transition.     LXVI. 

Resolution  vs.  Progression.     XXV,  XXVIII. 

Retardation.     LI,  LII,  LX. 

Rhythm.     LXIV,  LXIX. 
S  .  .Secondary  7th  Chords.     XXXV. 

Secondary  gth  Chords.     L,  LXVII. 

Secondary  Resolutions.     XXV,  XL,  XLVI,  LIV. 

Section.     LXVIII. 

Semi-phrase.     LXVIII. 

Sequence.    LXIII.    Also  Exs.  347,  349,  380,  522. 

Single  Tones  as  Chord  Representatives.     LV. 

Skips  of  a  3d  in  the  Melody.     X. 

Skips  of  a  4th  in  the  Melody.     XI. 

Sonata,  Sonatina.     LXX. 

Stationary  Tone :  Object  and  effect.     LVII. 

Sub-dominant  Harmony.     XLI,  XLII,  XLVII. 

Sub-tonic  as  a  Scale  Degree.     XLII,  XLIX. 

Supposed  Inharmonious  Progressions.     Exs.  541,  542. 

Suspension.     LI,  LII,  LX.     Also  Exs.  869  to  873. 
T  .  .Tetrachord.     I,  XLIX. 

Thorough-Base.     XLVIII. 

Tonality.     XVII,  XVIII,  XX,  XXXIII,  XXXVII,  XLIV,  LXVI,  LXX. 

Tone-quality  considered.     LXI,  LXIV.     Exs.  771,  772,  814. 

Timbre.     (See  Tone-quality.) 

Transition.     XXXII,  XXXVII,  LIV,  LXVI. 

Tritone.     LXII. 
U  .  .Unity  of  Design.     LXX. 

Unrulable  Progressions.     XII,  XXVIII,  XLVII,  LX,  LXII. 
V  .  .Vocal  Quartet.     LVIII,  LIX,  LX. 

Voice-parts.     LVIII  and  Preface. 

This  Index  is  designed  more  particularly  for  the  use  of  advanced  students 
in  reviewing  their  theoretical  work. 

The  subjects  treated  are  so  numerous  that  a  process  of  summarization  be- 
comes necessary.  All  the  information  upon  a  given  topic  must,  in  reviewing, 
be  gleaned  from  different  parts  of  the  book  and  focused  upon  that  particular 
point. 

This  will  also  enable  one  to  make  the  necessary  distinctions  between 
elementary  restrictions  and  final  applications. 

THE  AUTHOR. 


GOODRICH'S  ANALYTICAL  HARMONY. 


KEY  TO  EXAMPLES. 


(FOR   SELF-INSTRUCTION.) 


NOTE. 


T 


HE  principal  seventh  chords  are  indicated  thus : 

I.     Dominant  yth ; 

II.     Diminished  yth ; 

III.     Leading-note  yth. 


Secondary  seventh  chords  are  marked  IV,  V,  and  sometimes  III. 
Augmented-sixth  chords  are  indicated  by  cardinal  numbers,  i, 

*.  3- 

Inversions  are  marked  (i),  (2),  (3),  in  place  of  the  old  thorough- 

ba^e  figures. 

The  cipher,  o,  indicates  a  harmonic  note  above,  or  a  root-note 
below. 

Solution  to  Ex.  72.    Chapter  IX. 


(V/ith  connecting  links  throughout.) 


The  chord  of  E-flat  would  be  equally  correct  at  these  places  + ,  but 
the  G-minor  chord  affords  more  variety. 


394 


Ex.  77- 


GOODRICH  S   ANALYTICAL    HARMONY. 


J     _u  i 

g      &>      &> — .- 


*=* 


a- 


J      J 


r-^^-^ 


*=e 


Skips  of  a  3d.    Chapter  X. 


Ex.81. 


g       p 


;^- 


Skips  of  a  4th.    Chapter  XI. 


Ex.  87. 


S^ 

-T2^&- 


1=1: 


i 


=*=*3 

-    •*•    -^r 


GOODRICH  S   ANALYTICAL    HARMONY. 


395 


The  chord  of  F^  minor  may  be  used  at  (a)  and  (c),  or  the  D-major 
chord  could  be  substituted  at  (b). 

Thirty  Harmonic  Progressions  in  B=flat.     Chapter  XIII. 


g- 


r-         r 

The  first  five  of  the  thirty  progressions  are  given,  as  an  indication 
of  the  manner  in  which  the  others  are  to  be  written.  These  include 
every  possible  progression  by  means  of  concords. 

Preparatory  Theme  Harmonized  in  Two  Ways. 

fcfe 


3, 


1 


Ex.  109. 


Jt & ,«_ 

SBfc 


J         J          |       | 


These  should  be  written  in  vSeveral  scales.    The  order  of  progression 
is  reversible. 


396 


GOODRICH'S  ANALYTICAL  HARMONY. 
Exercise  on  a  Fundamental  Base. 


Ex.  117. 


:g— 3=:? 


*^f-9- 


-&• 


2T 


3^%=\ 

1      I  g     J 


(Two  re-arrangements  of  this.) 

Another  Method  for  Harmonizing  Skips  of  a  3d. 


Ex.  125. 


5858 


-O. ,«- 


a 


$: 


IMsconuected  l'roKr«-s-.i<»ii-». 


1     i^      g— ^-n 


^ 


BB 


f^ 


F=?s 


^~ 


fc 


2ig 


The  student's  exercise  should  correspond  exactly  to  this. 

Theme  Harmonized  in  A-minor. 
Ex.  132. 


2 


2=: 


Ex.  141. 


C-major  and  Relative  Minor  Combined. 


^      m    ^      ^rra  .  \  •    m     -  m  \  >g'-5-r-g-ik-aLr*'-^i    •  \  •      \ — ^  i   J       i 


-*— ^ 


J  *  L  r 


GOODRICH  S    ANALYTICAL    HARMONY. 

Chapter  XVIII.    Table  of  Modulations. 


397 


r=^ 


'&=^%- 


w^s- 


RH 


to  C  min.          BJ2  to  D  min.          BJ2  to  F." 


BJ2  to  G  min. 


Two  more  arrangements  of  this,  with  the  same  base. 
Modulatory  Theme  Harmonized. 


Ex.  154. 


T2==4=q= 

r^— =1— — -S^=g *-+!$— 

kg , jj—  M \-*<& • 0- 

l ^ 1 ^ 1——& * 


Concords  only. 


Efc 


a^j--i=^=jj^E^=^ 


EEE£ 


3d  or  sth  Omitted  from  Dominant  yth  Chords. 


Ex.  185. 


m 


Major  and  Minor  Resolutions. 


Ex.  193. 


tdb 


if,    *       »- — >       £H-iT i" 


:fr: 


-» 1 ^ 


=t±=ib=Jzd?Q 
3EM^^=^=3 


^ 


O          I 


398 


GOODKICH'S  ANALYTICAL  HARMONY. 


=fe: 


-1 0- 


Inverted  Bases.    Chapter  XXVII. 


Ex.  248. 


HH 


2?       ^J 


(2)  (2)          (2)  (2)  (2)     o      i 

The  inversions  (2)  are  the  result  of  melodic  progressions  in  the  base. 

Primary  Resolutions  of  the  Diminished  jth. 


Ex.  306. 


AH  P"  *-^^*  *  

—  0  0  ^  — 

P         *         f        0 

4     '    P 

t^ssstf.          p 

V-y    •  r        o        0        j 

*.__ 

i        n                                                               i        ii 

cy  1  *  —  Jp  f  — 

.      -4 

_/  —  -9  — 

-^  —  ,  —  ^*  —  0— 

r 


i      ii 


fE 


Ex.  309. 


y&-^2- 


AH  fundamental  progressions,  excepting  + ,  (2). 

Corresponding  Dominant  7th  Chords.    Chapter  XXXII. 

rffetF 


i 


Ti"  i  * 


>   -x    '      #J#       '      I    >     ,         ^2~ 

-  ff^SB3rr?Mi*S 


Enharmonic  equivalents,  (a)  and  (b). 


GOODRICH'S  ANALYTICAL  HARMONY.  399 

Intermediate  and  Terminal  Resolutions.     Chapter  XXXIII. 


%=3=£=&&^±-U&^=f=*=\ 

-0 -m 9^- m 1 9 m ^^ m 1 t"          ^ " " 1 

._ — » — _ — 1_1  _i 1 , ! 3 


f^-f-rf-M* 


-+ » 


\ 


(1)    (1)    (2)     o          (1)    (1)    (2)     o 
+4*1    j      J-J 


1 


(2)    (1)    (2)     0 
Two  more  arrangements  of  the  last. 


3200 


335- 


Farther  View  of  Inverted  Bases. 

^-4 -'  -  ' 


-J— J-jtLJ iT-^— -4— 1-+^.   i-3*—*- 


^ 


;4r 


l=±=: 
>      •      ^^ 


3E 


(2) 


(2)      (1)     (2) 


(2) 


-» — * — i- 


«=^=3? 


1 


(2)  (3)          i       -*• 

*  The  yth  ascends  here  to  avoid  the  half-open  position,  and  because 
this  is  not  a  final  cadence.    It  is,  rather,  an  intermediate  progression. 

Chapter  XXXVI.    Exs.  365  and  366. 

In  the  first  chord  g  and  b  represent  the  G-major  triad.    Therelj/e 
d  is  the  remaining  note.    In  the  next  two  chords  e  is  wanting,  thus  : 


400 


GOODRICH'S  ANALYTICAL  HARMONY. 


The  fourth  and  fifth  chords  are  complete.     Small  notes  indicate  th. 
remaining  intervals  of  the  sixth,  seventh  and  eighth  chords : 


X 


The  harmony  is  sufficiently  represented  in  the  original  example. 
But  in  supplying  adventitious  parts  it  is  often  necessary  to  under- 
stand the  theory  of  chord  representation,  especially  where  but  two 
notes  of  a  four-fold  discord  are  present.  It  is  evident  that  d  is  want- 
ing in  the  last  two  chords  of  Ex.  366. 

Sequence  of  Essential  Discords. 


Ex.  380. 


I  I  I  I  I 

II  I  I  I  O 


(2) 
Chapter  XXXVIII. 

Enharmonic  equivalents. 


GOODRICH'S  ANALYTICAL,  HARMONY. 


401 


These  represent  merely  the  changes  in  notation  of  the  three  primr.ry 
diminished  yth  chords,  A,  B  and  C,  resolving  naturally  to  fifteen 
minor  key-notes.  The  inversions  are  also  to  be  worked  out. 

Sequence  of  Diminished  yth  Chords. 


Ex.  408. 


Ex.  429. 


Passing  Chords.    Chapter  XL. 


«-  — & fi—a 


*— rr 


Ex,  481 
and  482. 


Augmented  6th  Chords. 

No.  1.  |         I 


Ex.   505.    \ 


3. 


rp3ZZC           "                       f 

»                                    0 

_            I*  _                        1 

-S     >i 

1 

^            Y  '                 ^J 

Re-arrange,  but  do  not  invert,  this  exercise. 


402 


Ei..  506. 


GOODRICH  S    ANALYTICAL    HARMONY. 


g 


— * « — -fa * **• 


>s 


in      2. 


|^i^__j  j. 

J- 

V-&  *_ 

0     1;    »     ^       f        * 

.  *, 

. 

2S3  —  T—     ^?~ 

Pu*  1*k£~ 

—  p  1Q0  — 

-  -2—  —  *  *—  —  -*  — 

--*  —  H 

KF         QF 

7» 

DM 

832        III 

'  r    ^r 

1 

-  i      '  *       f      "r 

(S'          1 

IV           ^ 

1 

i     i     i     r 

1 

g^  T   r   J 

—I/  —  [?«(- 

—  1  1— 

p-^-f  ^ 

^=\ 

(2) 

Sequence  of  Dominant  jth  Chords  Terminating  with  an  Augmented 
6th  Chord  for  the  Cadence.    Chapter  XLVI. 


~**m  iTu     l,r  ?"»•      •       <2 ijJ       e>  • 


b      In  minor. 


In  example  (a)  the  chord  numbered  3  is  an  enharmonic  expediency ; 
c%  and  <?Q  being  substituted  for  db  and/>  of  the  essential  discord  on 
G-flat. 

In  example  (b)  only  one  enharmonic  change  is  made :  C  ?  is 
substituted  for  d  t>  in  the  descending  sequence  in  order  to  make  the 
direct  cadence  in  G-minor.  In  all  such  instances  the  augmented  6th 
chords  are  resolved  according  to  previous  directions. 

Suspensions  Resolving  to  a  Changing  Harmony. 


Ex.  595. 


fr-}     0 

&       (S 

,5.  S 

-fi>:  &  1 

t^-r           "^    r      —  ^^^f" 

I^-W- 

-= 

GOODRICH'S  ANALYTICAL  HARMONY. 
Appoggiaturas  Harmonized. 


403 


Ex.  669. 


„  20         p*. 

foff        PTJ        I      *—i- 

i=tztzz==  =|= 


Harmonic  Counterpoint.    Chapter  LIX. 

2  3 


Ex.  729. 


*  In  place  of  this  passing  diminished  yth  chord  the  essential  discord 
may  be  retained,  considering  the  <:#  as  an  appoggiatura. 


Passing  Tones  and  Appoggiaturas. 


Ex.  73irt. 

M 


In  the  first  measure  the  tenor  might  sing  the  tone  between  a  and  c, 
as  it  would  accord  very  well  with  the  base  and  soprano.  But  this 
would  compel  the  contralto  to  descend  to  e.  A  slightly  different 
arrangement  is  added 


Ex.  732^. 


404 


GOODRICK'S  ANALYTICAL  HARMONY. 


This  is  an  improvement,  though  the  first  measure  is  really  melodic 
counterpoint. 

Elaboration  of  Ex.  739. 


3  i 


t=f=? 


I     LJ 
Harmonic  Sequence.    Chapter  I. XIII. 


Ex.  8oga. 


•      5 

j-*      « 

__  \       | 

1 

fm  ff 

* 

i      P 

w 

0           ~1 

"  K 

i        ^ 

^35- 

S        S 

4      J 

J 

«/ 

«- 

1        F 

3d. 

m/*~~  m 

*     ^ 

4th. 

3d. 

^ 

i  i 

.^. 

^ 

t~\*  ^ 

J-—            1 

*'           tf 

i      1 

**  ^  _ 

• 

«      » 

^ 

^^     '  ^ 

i 

2: 

IT 

>J 

r  1 

0 

J 

kj 

Lj 

-*• 

fix.  8ogi>. 


Secondary  and  Principal  Sevenths. 


is* 


^ '  i^-^-h-11— 1 

; — g~r^ —  -f-g       '- 

zz±  =tg— ^Z=*±n^:} 


'     b 
V  V 


III  IV 


(3)    (1) 


(1) 


£^^^^ 
(3)     (1J 


Motives  for  Sonatinas  in  F-major  and  in  D-minor- 

Allpgrro.  a 


niod«*rato.       b 


Each  motive  is  divided  into  semi-phrases,  which  may  be  used  sepa- 
rately in  the  development. 


